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\ihead{The eta-basis of QSym}
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\begin{document}

\title{The eta-basis of $\operatorname*{QSym}$ (DRAFT)}
\author{Darij Grinberg}
\date{
%TCIMACRO{\TeXButton{today}{\today}}%
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\maketitle
\tableofcontents

\section*{***}

This is just a skeleton of what will hopefully become a paper.

The paper will construct a new family $\left(  \eta_{\alpha}\right)
_{\alpha\in\operatorname*{Comp}}$ of quasisymmetric functions that is a basis
of $\operatorname*{QSym}$ whenever $2$ is invertible in the base ring. We will
show a formula for expanding products of the form $\eta_{\alpha}\eta_{\beta}$
as $\mathbb{Z}$-linear combinations of $\eta_{\gamma}$'s, and we will apply it
to partially solve \cite[Question 2.51]{Grinbe18}.

Once again: \textbf{this is nowhere near finished} (partially more of a todo
list than a paper).

\subsubsection*{Acknowledgments}

I thank Marcelo Aguiar, G\'{e}rard H. E. Duchamp, Angela Hicks, Vasu Tewari,
Katya Vassilieva, Alexander Zhang, and Yan Zhuang for interesting and helpful
conversations. The SageMath computer algebra system \cite{SageMath} has been
used in discovering some of the results.

I am grateful to Sara Billey, Petter Br\"{a}nd\'{e}n, Sylvie Corteel, and
Svante Linusson for organizing the Spring Semester 2020 in Algebraic and
Enumerative Combinatorics at the Institut Mittag-Leffler, at which much of
this paper has been written.

\begin{fineprint}
This material is based upon work supported by the Swedish Research Council
under grant no. 2016-06596 while the author was in residence at Institut
Mittag-Leffler in Djursholm, Sweden during Spring 2020.
\end{fineprint}

\section{Introduction}

\subsection{Formal power series and quasisymmetry}

We will use some of the standard notations from \cite[Chapter 5]{GriRei}. Namely:

\begin{itemize}
\item We let $\mathbb{N}=\left\{  0,1,2,\ldots\right\}  $.

\item We fix a commutative ring $\mathbf{k}$.

\item We consider the ring $\mathbf{k}\left[  \left[  x_{1},x_{2},x_{3}%
,\ldots\right]  \right]  $ of formal power series in countably many commuting
variables $x_{1},x_{2},x_{3},\ldots$. A \emph{monomial} shall mean a formal
expression of the form $x_{1}^{\alpha_{1}}x_{2}^{\alpha_{2}}x_{3}^{\alpha_{3}%
}\cdots$, where $\alpha=\left(  \alpha_{1},\alpha_{2},\alpha_{3}%
,\ldots\right)  \in\mathbb{N}^{\infty}$ is a sequence of nonnegative integers
such that only finitely many $\alpha_{i}$ are positive. Formal power series
are formal infinite $\mathbf{k}$-linear combinations of such monomials.

\item Each monomial $x_{1}^{\alpha_{1}}x_{2}^{\alpha_{2}}x_{3}^{\alpha_{3}%
}\cdots$ has degree $\alpha_{1}+\alpha_{2}+\alpha_{3}+\cdots$.

\item A formal power series $f\in\mathbf{k}\left[  \left[  x_{1},x_{2}%
,x_{3},\ldots\right]  \right]  $ is said to be \emph{of bounded degree} if
there exists some $d\in\mathbb{N}$ such that each monomial in $f$ has degree
$\leq d$ (that is, each monomial of degree $>d$ has coefficient $0$ in $f$).
\end{itemize}

For example, the formal power series $\left(  x_{1}+x_{2}+x_{3}+\cdots\right)
^{3}$ is of bounded degree, but the formal power series $\dfrac{1}{1-x_{1}%
}=1+x_{1}+x_{1}^{2}+x_{1}^{3}+\cdots$ is not.

\textbf{Question:} What do the monomials $x_{1}^{4}x_{3}^{7}x_{4}x_{9}^{2}$
and $x_{3}^{4}x_{4}^{7}x_{10}x_{16}^{2}$ and $x_{5}^{4}x_{6}^{7}x_{8}x_{9}%
^{2}$ have in common (but not in common with $x_{1}^{7}x_{3}^{4}x_{4}x_{9}%
^{2}$) ?

\textbf{Answer:} They have the same sequence of nonzero exponents (when the
variables are ordered in increasing order -- i.e., if $i<j$, then $x_{i}$ goes
before $x_{j}$). Or, to put it differently, they all have the form $x_{a}%
^{4}x_{b}^{7}x_{c}x_{d}^{2}$ for some $a<b<c<d$. We shall call such monomials
\textit{pack-equivalent}.

Let us define this concept more rigorously:

\begin{definition}
Two monomials $\mathfrak{m}$ and $\mathfrak{n}$ are said to be
\emph{pack-equivalent} if they can be written in the forms%
\[
\mathfrak{m}=x_{i_{1}}^{a_{1}}x_{i_{2}}^{a_{2}}\cdots x_{i_{\ell}}^{a_{\ell}%
}\ \ \ \ \ \ \ \ \ \ \text{and}\ \ \ \ \ \ \ \ \ \ \mathfrak{n}=x_{j_{1}%
}^{a_{1}}x_{j_{2}}^{a_{2}}\cdots x_{j_{\ell}}^{a_{\ell}}%
\]
for some $\ell\in\mathbb{N}$, some positive integers $a_{1},a_{2}%
,\ldots,a_{\ell}$ and two strictly increasing $\ell$-tuples $\left(
i_{1}<i_{2}<\cdots<i_{\ell}\right)  $ and $\left(  j_{1}<j_{2}<\cdots<j_{\ell
}\right)  $ of positive integers.
\end{definition}

For example, the monomials $x_{1}^{4}x_{3}^{7}x_{4}x_{9}^{2}$ and $x_{3}%
^{4}x_{4}^{7}x_{10}x_{16}^{2}$ are pack-equivalent, since they can be written
as $x_{1}^{4}x_{3}^{7}x_{4}x_{9}^{2}=x_{i_{1}}^{a_{1}}x_{i_{2}}^{a_{2}}\cdots
x_{i_{\ell}}^{a_{\ell}}$ and $x_{3}^{4}x_{4}^{7}x_{10}x_{16}^{2}=x_{j_{1}%
}^{a_{1}}x_{j_{2}}^{a_{2}}\cdots x_{j_{\ell}}^{a_{\ell}}$ for $\ell=4$ and
$\left(  a_{1},a_{2},\ldots,a_{\ell}\right)  =\left(  4,7,1,2\right)  $ and
$\left(  i_{1}<i_{2}<\cdots<i_{\ell}\right)  =\left(  1,3,4,9\right)  $ and
$\left(  j_{1}<j_{2}<\cdots<j_{\ell}\right)  =\left(  3,4,10,16\right)  $.

\begin{definition}
\textbf{(a)} A formal power series $f\in\mathbf{k}\left[  \left[  x_{1}%
,x_{2},x_{3},\ldots\right]  \right]  $ is \emph{quasisymmetric} if it has the
property that any two pack-equivalent monomials have the same coefficient in
$f$ (that is: if $\mathfrak{m}$ and $\mathfrak{n}$ are two pack-equivalent
monomials, then the coefficient of $\mathfrak{m}$ in $f$ equals the
coefficient of $\mathfrak{n}$ in $f$).

\textbf{(b)} A \emph{quasisymmetric function} means a formal power series
$f\in\mathbf{k}\left[  \left[  x_{1},x_{2},x_{3},\ldots\right]  \right]  $
that is quasisymmetric and of bounded degree.
\end{definition}

Quasisymmetric functions are studied in \cite[Chapters 5--6]{GriRei},
\cite[\S 7.19]{Stanley-EC2}, \cite[Chapter 8]{Sagan20} and elsewhere.

It is known (\cite[Proposition 5.1.3]{GriRei}) that the set of all
quasisymmetric functions is a $\mathbf{k}$-subalgebra of $\mathbf{k}\left[
\left[  x_{1},x_{2},x_{3},\ldots\right]  \right]  $. It is denoted by
$\operatorname*{QSym}$ and called the \emph{ring of quasisymmetric functions}.
It has several bases (as a $\mathbf{k}$-module), most of which are indexed by compositions.

\subsection{Compositions}

A \emph{composition} means a finite list $\left(  \alpha_{1},\alpha_{2}%
,\ldots,\alpha_{k}\right)  $ of positive integers. The set of all compositions
will be denoted by $\operatorname*{Comp}$. The \emph{empty composition
}$\varnothing$ is the composition $\left(  {}\right)  $, which is a $0$-tuple.

The \emph{length }$\ell\left(  \alpha\right)  $ of a composition
$\alpha=\left(  \alpha_{1},\alpha_{2},\ldots,\alpha_{k}\right)  $ is defined
to be the number $k$.

If $\alpha=\left(  \alpha_{1},\alpha_{2},\ldots,\alpha_{k}\right)  $ is a
composition, then the nonnegative integer $\alpha_{1}+\alpha_{2}+\cdots
+\alpha_{k}$ is called the \emph{size} of $\alpha$ and is denoted by
$\left\vert \alpha\right\vert $. For any $n\in\mathbb{N}$, we define a
\emph{composition of }$n$ to be a composition that has size $n$. We let
$\operatorname*{Comp}\nolimits_{n}$ be the set of all compositions of $n$ (for
given $n\in\mathbb{N}$). The notation \textquotedblleft$\alpha\models
n$\textquotedblright\ is short for \textquotedblleft$\alpha\in
\operatorname*{Comp}\nolimits_{n}$\textquotedblright. For example, $\left(
1,5,2,1\right)  $ is a composition with size $9$, so that $\left\vert \left(
1,5,2,1\right)  \right\vert =1+5+2+1=9$, so that $\left(  1,5,2,1\right)
\in\operatorname*{Comp}\nolimits_{9}$, or, in other words, $\left(
1,5,2,1\right)  \models9$.

For any $n\in\mathbb{Z}$, we let $\left[  n\right]  $ denote the set $\left\{
1,2,\ldots,n\right\}  $. This set is empty whenever $n\leq0$, and otherwise
has size $n$.

It is well-known that any positive integer $n$ has exactly $2^{n-1}$
compositions. This has a standard bijective proof (\textquotedblleft stars and
bars\textquotedblright) which is worth recalling in detail, as the bijection
itself will be used a lot:

\begin{definition}
Let $n\in\mathbb{N}$. Let $\mathcal{P}\left(  \left[  n-1\right]  \right)  $
be the powerset of $\left[  n-1\right]  $ (that is, the set of all subsets of
$\left[  n-1\right]  $).

\textbf{(a)} We define a map $D:\operatorname*{Comp}\nolimits_{n}%
\rightarrow\mathcal{P}\left(  \left[  n-1\right]  \right)  $ by
\begin{align*}
D\left(  \alpha_{1},\alpha_{2},\ldots,\alpha_{k}\right)   &  =\left\{
\alpha_{1}+\alpha_{2}+\cdots+\alpha_{i}\ \mid\ i\in\left[  k-1\right]
\right\} \\
&  =\left\{  \alpha_{1}<\alpha_{1}+\alpha_{2}<\alpha_{1}+\alpha_{2}+\alpha
_{3}<\cdots<\alpha_{1}+\alpha_{2}+\cdots+\alpha_{k-1}\right\}  .
\end{align*}


\textbf{(b)} We define a map $\operatorname*{comp}:\mathcal{P}\left(  \left[
n-1\right]  \right)  \rightarrow\operatorname*{Comp}\nolimits_{n}$ as follows:
For any $I\in\mathcal{P}\left(  \left[  n-1\right]  \right)  $, we set%
\[
\operatorname*{comp}\left(  I\right)  =\left(  i_{1}-i_{0},i_{2}-i_{1}%
,\ldots,i_{m}-i_{m-1}\right)  ,
\]
where $i_{0},i_{1},\ldots,i_{m}$ are the elements of the set $I\cup\left\{
0,n\right\}  $ in increasing order (so that $i_{0}<i_{1}<\cdots<i_{m}$,
therefore $i_{0}=0$ and $i_{m}=n$ and $\left\{  i_{1}<i_{2}<\cdots
<i_{m-1}\right\}  =I$).

The maps $D$ and $\operatorname*{comp}$ are mutually inverse bijections. (See
\cite[detailed version, Proposition 10.17]{Grinbe15} for a detailed proof of this.)
\end{definition}

The notation $D$ we just introduced presumably originates in the word
\textquotedblleft\textbf{d}escent\textquotedblright, but the connection
between $D$ and actual descents is indirect and rather misleading. I prefer to
call $D$ the \textquotedblleft partial sum map\textquotedblright\ (as
$D\left(  \alpha\right)  $ consists of the partial sums of the composition
$\alpha$) and its inverse $\operatorname*{comp}$ the \textquotedblleft
interstitial map\textquotedblright\ (as $\operatorname*{comp}\left(  I\right)
$ consists of the lengths of the intervals into which the elements of $I$
split the interval $\left[  n\right]  $).

Note that every composition $\alpha$ of size $\left\vert \alpha\right\vert >0$
satisfies $\left\vert D\left(  \alpha\right)  \right\vert =\ell\left(
\alpha\right)  -1$, so that $\left\vert D\left(  \alpha\right)  \right\vert
+1=\ell\left(  \alpha\right)  $. But this fails if $\alpha$ is the empty
composition $\varnothing=\left(  {}\right)  $ (since $D\left(  {}\right)
=\varnothing$ and $\ell\left(  {}\right)  =0$).

\subsection{The monomial and fundamental bases of $\operatorname*{QSym}$}

We will only need two bases of $\operatorname*{QSym}$: the monomial basis and
the fundamental basis.

If $\alpha=\left(  \alpha_{1},\alpha_{2},\ldots,\alpha_{\ell}\right)  $ is a
composition, then we define the \emph{monomial quasisymmetric function}
$M_{\alpha}\in\operatorname*{QSym}$ by%
\begin{equation}
M_{\alpha}=\sum_{i_{1}<i_{2}<\cdots<i_{\ell}}x_{i_{1}}^{\alpha_{1}}x_{i_{2}%
}^{\alpha_{2}}\cdots x_{i_{\ell}}^{\alpha_{\ell}}=\sum_{\substack{\mathfrak{m}%
\text{ is a monomial pack-equivalent}\\\text{to }x_{1}^{\alpha_{1}}%
x_{2}^{\alpha_{2}}\cdots x_{\ell}^{\alpha_{\ell}}}}\mathfrak{m}.
\label{eq.Malpha.def}%
\end{equation}


For example,%
\[
M_{\left(  2,1\right)  }=\sum_{i<j}x_{i}^{2}x_{j}=x_{1}^{2}x_{2}+x_{1}%
^{2}x_{3}+x_{2}^{2}x_{3}+x_{1}^{2}x_{4}+x_{2}^{2}x_{4}+x_{3}^{2}x_{4}+\cdots.
\]


The family $\left(  M_{\alpha}\right)  _{\alpha\in\operatorname*{Comp}}$ is a
basis of the $\mathbf{k}$-module $\operatorname*{QSym}$, and is known as the
\emph{monomial basis} of $\operatorname*{QSym}$.

For any composition $\alpha$, we define the \emph{fundamental quasisymmetric
function }$L_{\alpha}\in\operatorname*{QSym}$ by
\begin{equation}
L_{\alpha}=\sum_{\substack{\beta\in\operatorname*{Comp}\nolimits_{n}%
;\\D\left(  \beta\right)  \supseteq D\left(  \alpha\right)  }}M_{\beta},
\label{eq.Lalpha.def}%
\end{equation}
where $n=\left\vert \alpha\right\vert $ (so that $\alpha\in
\operatorname*{Comp}\nolimits_{n}$). It is not hard to rewrite this as%
\begin{equation}
L_{\alpha}=\sum_{\substack{i_{1}\leq i_{2}\leq\cdots\leq i_{n};\\i_{j}%
<i_{j+1}\text{ whenever }j\in D\left(  \alpha\right)  }}x_{i_{1}}x_{i_{2}%
}\cdots x_{i_{n}} \label{eq.Lalpha.xprod}%
\end{equation}
(again with $n=\left\vert \alpha\right\vert $). This quasisymmetric function
$L_{\alpha}$ is sometimes denoted by $F_{\alpha}$.

The family $\left(  L_{\alpha}\right)  _{\alpha\in\operatorname*{Comp}}$ is a
basis of the $\mathbf{k}$-module $\operatorname*{QSym}$, and is known as the
\emph{fundamental basis} of $\operatorname*{QSym}$.

Using M\"{o}bius inversion on the Boolean lattice $\mathcal{P}\left(  \left[
n-1\right]  \right)  $, the definition (\ref{eq.Lalpha.def}) of the
fundamental basis can be turned around to obtain an expression of the
$M_{\alpha}$ in the fundamental basis. Namely, if $\alpha$ is a composition,
and if $n=\left\vert \alpha\right\vert $, then%
\[
M_{\alpha}=\sum_{\substack{\beta\in\operatorname*{Comp}\nolimits_{n}%
;\\D\left(  \beta\right)  \supseteq D\left(  \alpha\right)  }}\left(
-1\right)  ^{\ell\left(  \beta\right)  -\ell\left(  \alpha\right)  }L_{\beta
}.
\]
(See \cite[Proposition 5.2.8]{GriRei} for more details of the proof. In a
nutshell, the equality follows from M\"{o}bius inversion using the fact that
$\left\vert D\left(  \beta\right)  \setminus D\left(  \alpha\right)
\right\vert =\ell\left(  \beta\right)  -\ell\left(  \alpha\right)  $ whenever
$\alpha,\beta\in\operatorname*{Comp}\nolimits_{n}$ satisfy $D\left(
\beta\right)  \supseteq D\left(  \alpha\right)  $.)

\section{The $\eta$-basis of $\operatorname*{QSym}$}

\subsection{The $\eta$-functions}

I shall now define a new family of quasisymmetric functions:

\begin{definition}
\label{def.etaalpha}For any $n\in\mathbb{N}$ and any composition $\alpha
\in\operatorname*{Comp}\nolimits_{n}$, we define a quasisymmetric function
$\eta_{\alpha}\in\operatorname*{QSym}$ by%
\begin{equation}
\eta_{\alpha}=\sum_{\substack{\beta\in\operatorname*{Comp}\nolimits_{n}%
;\\D\left(  \beta\right)  \subseteq D\left(  \alpha\right)  }}2^{\ell\left(
\beta\right)  }M_{\beta}. \label{eq.def.etaalpha.def}%
\end{equation}

\end{definition}

\begin{example}
\textbf{(a)} Setting $n=5$ and $\alpha=\left(  1,3,1\right)  $ in this
definition, we obtain%
\begin{align*}
\eta_{\left(  1,3,1\right)  }  &  =\sum_{\substack{\beta\in
\operatorname*{Comp}\nolimits_{5};\\D\left(  \beta\right)  \subseteq D\left(
1,3,1\right)  }}2^{\ell\left(  \beta\right)  }M_{\beta}=\sum_{\substack{\beta
\in\operatorname*{Comp}\nolimits_{5};\\D\left(  \beta\right)  \subseteq
\left\{  1,4\right\}  }}2^{\ell\left(  \beta\right)  }M_{\beta}%
\ \ \ \ \ \ \ \ \ \ \left(  \text{since }D\left(  1,3,1\right)  =\left\{
1,4\right\}  \right) \\
&  =2^{\ell\left(  5\right)  }M_{\left(  5\right)  }+2^{\ell\left(
1,4\right)  }M_{\left(  1,4\right)  }+2^{\ell\left(  4,1\right)  }M_{\left(
4,1\right)  }+2^{\ell\left(  1,3,1\right)  }M_{\left(  1,3,1\right)  }%
\end{align*}
(since the compositions $\beta\in\operatorname*{Comp}\nolimits_{5}$ satisfying
$D\left(  \beta\right)  \subseteq\left\{  1,4\right\}  $ are $\left(
5\right)  $, $\left(  1,4\right)  $, $\left(  4,1\right)  $ and $\left(
1,3,1\right)  $). This simplifies to%
\[
\eta_{\left(  1,3,1\right)  }=2M_{\left(  5\right)  }+4M_{\left(  1,4\right)
}+4M_{\left(  4,1\right)  }+8M_{\left(  1,3,1\right)  }.
\]


\textbf{(b)} For any positive integer $n$, we have $\eta_{\left(  n\right)
}=2M_{\left(  n\right)  }$, because the only composition $\beta\in
\operatorname*{Comp}\nolimits_{n}$ satisfying $D\left(  \beta\right)
\subseteq D\left(  n\right)  $ is the composition $\left(  n\right)  $ itself
(since $D\left(  n\right)  $ is the empty set $\varnothing$) and has length
$\ell\left(  n\right)  =1$. Likewise, the empty composition $\varnothing
=\left(  {}\right)  $ satisfies $\eta_{\varnothing}=M_{\varnothing}$.
\end{example}

When $\alpha$ is an odd composition (i.e., all entries of $\alpha$ are odd),
our definition of $\eta_{\alpha}$ is precisely the one given in \cite[(6.1)]%
{AgBeSo14}, and differs only in sign from the one given in \cite[(2.1)]%
{Hsiao07} (because of \cite[Proposition 2.1]{Hsiao07}). Our main innovation is
extending this definition to arbitrary compositions $\alpha$.

The following is easy to see:

\begin{proposition}
\label{prop.eta.through-x}Let $n\in\mathbb{N}$ and $\alpha\in
\operatorname*{Comp}\nolimits_{n}$. Then,%
\[
\eta_{\alpha}=\sum_{\substack{g=\left(  g_{1}\leq g_{2}\leq\cdots\leq
g_{n}\right)  ;\\g_{i}=g_{i+1}\text{ for each }i\in\left[  n-1\right]
\setminus D\left(  \alpha\right)  }}2^{\left\vert \left\{  g_{1},g_{2}%
,\ldots,g_{n}\right\}  \right\vert }x_{g_{1}}x_{g_{2}}\cdots x_{g_{n}},
\]
where the sum is over all weakly increasing $n$-tuples $g=\left(  g_{1}\leq
g_{2}\leq\cdots\leq g_{n}\right)  $ of positive integers that satisfy $\left(
g_{i}=g_{i+1}\text{ for each }i\in\left[  n-1\right]  \setminus D\left(
\alpha\right)  \right)  $.
\end{proposition}

\begin{proof}
TODO. The slickest way to prove this is using the definition of $\eta_{\alpha
}$ and \cite[detailed version, Proposition 10.10]{Grinbe15}.
\end{proof}

\begin{proposition}
\label{prop.eta.through-x2}Let $\alpha=\left(  \alpha_{1},\alpha_{2}%
,\ldots,\alpha_{k}\right)  \in\operatorname*{Comp}$. Then,%
\[
\eta_{\alpha}=\sum_{h_{1}\leq h_{2}\leq\cdots\leq h_{k}}2^{\left\vert \left\{
h_{1},h_{2},\ldots,h_{k}\right\}  \right\vert }x_{h_{1}}^{\alpha_{1}}x_{h_{2}%
}^{\alpha_{2}}\cdots x_{h_{k}}^{\alpha_{k}}.
\]

\end{proposition}

\begin{proof}
TODO. The slickest way to prove this is by imitating \cite[detailed version,
Proposition 10.69]{Grinbe15}, after realizing that Proposition
\ref{prop.eta.through-x} rewrites as%
\[
\eta_{\alpha}=\sum_{\substack{g=\left(  g_{1}\leq g_{2}\leq\cdots\leq
g_{n}\right)  ;\\\left\{  j\in\left[  n-1\right]  \ \mid\ g_{j}<g_{j+1}%
\right\}  \subseteq D\left(  \alpha\right)  }}2^{\left\vert \left\{
g_{1},g_{2},\ldots,g_{n}\right\}  \right\vert }x_{g_{1}}x_{g_{2}}\cdots
x_{g_{n}}.
\]

\end{proof}

We can also write the $\eta_{\alpha}$ in the fundamental basis:

\begin{proposition}
\label{prop.eta.through-F}Let $n$ be a positive integer. Let $\alpha
\in\operatorname*{Comp}\nolimits_{n}$. Then,%
\[
\eta_{\alpha}=2\sum_{\gamma\in\operatorname*{Comp}\nolimits_{n}}\left(
-1\right)  ^{\left\vert D\left(  \gamma\right)  \setminus D\left(
\alpha\right)  \right\vert }L_{\gamma}.
\]

\end{proposition}

This proposition generalizes \cite[Proposition 2.2]{Hsiao07}, and is a bit
similar to the discrete Radon transform on a hypercube (\cite[ca. Theorem
2.2]{Stanley-AC}).

It can be proved using the following simple binomial identity:

\begin{lemma}
\label{lem.eta.through-F.lem1}Let $S$ and $T$ be two finite sets. Then,%
\[
\sum_{I\subseteq S}\left(  -1\right)  ^{\left\vert I\setminus T\right\vert }=%
\begin{cases}
2^{\left\vert S\right\vert }, & \text{if }S\subseteq T;\\
0, & \text{otherwise}%
\end{cases}
\ \ .
\]

\end{lemma}

\begin{proof}
[Proof of Lemma \ref{lem.eta.through-F.lem1}.]Here is a sketch; a detailed
proof can be found in \cite[solution to Exercise 2.9.11]{Grinbe20}.

If $S\subseteq T$, then each subset $I$ of $S$ satisfies $I\setminus
T=\varnothing$, and thus the sum $\sum_{I\subseteq S}\left(  -1\right)
^{\left\vert I\setminus T\right\vert }$ is a sum of $2^{\left\vert
S\right\vert }$ many addends each equal to $\left(  -1\right)  ^{\left\vert
\varnothing\right\vert }=1$. On the other hand, if $S\not \subseteq T$, then
there exists some $v\in S$ such that $v\notin T$, and therefore the addends of
the sum $\sum_{I\subseteq S}\left(  -1\right)  ^{\left\vert I\setminus
T\right\vert }$ cancel each other out in pairs (viz., for each subset $K$ of
$S\setminus\left\{  v\right\}  $, the addend for $I=K$ cancels the addend for
$I=K\cup\left\{  v\right\}  $); thus, the sum is $0$ in this case.
\end{proof}

\begin{proof}
[Proof of Proposition \ref{prop.eta.through-F}.]We have%
\begin{align*}
&  2\sum_{\gamma\in\operatorname*{Comp}\nolimits_{n}}\left(  -1\right)
^{\left\vert D\left(  \gamma\right)  \setminus D\left(  \alpha\right)
\right\vert }\underbrace{L_{\gamma}}_{\substack{=\sum_{\substack{\beta
\in\operatorname*{Comp}\nolimits_{n};\\D\left(  \beta\right)  \supseteq
D\left(  \gamma\right)  }}M_{\beta}\\\text{(by the definition of }L_{\gamma
}\text{)}}}\\
&  =2\sum_{\gamma\in\operatorname*{Comp}\nolimits_{n}}\left(  -1\right)
^{\left\vert D\left(  \gamma\right)  \setminus D\left(  \alpha\right)
\right\vert }\sum_{\substack{\beta\in\operatorname*{Comp}\nolimits_{n}%
;\\D\left(  \beta\right)  \supseteq D\left(  \gamma\right)  }}M_{\beta}\\
&  =2\sum_{\beta\in\operatorname*{Comp}\nolimits_{n}}\sum_{\substack{\gamma
\in\operatorname*{Comp}\nolimits_{n};\\D\left(  \beta\right)  \supseteq
D\left(  \gamma\right)  }}\left(  -1\right)  ^{\left\vert D\left(
\gamma\right)  \setminus D\left(  \alpha\right)  \right\vert }M_{\beta}.
\end{align*}
But every $\beta\in\operatorname*{Comp}\nolimits_{n}$ satisfies%
\begin{align*}
\sum_{\substack{\gamma\in\operatorname*{Comp}\nolimits_{n};\\D\left(
\beta\right)  \supseteq D\left(  \gamma\right)  }}\left(  -1\right)
^{\left\vert D\left(  \gamma\right)  \setminus D\left(  \alpha\right)
\right\vert }  &  =\sum_{\substack{I\subseteq\left[  n-1\right]  ;\\D\left(
\beta\right)  \supseteq I}}\left(  -1\right)  ^{\left\vert I\setminus D\left(
\alpha\right)  \right\vert }\\
&  \ \ \ \ \ \ \ \ \ \ \left(
\begin{array}
[c]{c}%
\text{here, we have substituted }I\text{ for }D\left(  \gamma\right)  \text{
in the sum,}\\
\text{since the map }D:\operatorname*{Comp}\nolimits_{n}\rightarrow
\mathcal{P}\left(  \left[  n-1\right]  \right)  \text{ is a bijection}%
\end{array}
\right) \\
&  =\sum_{I\subseteq D\left(  \beta\right)  }\left(  -1\right)  ^{\left\vert
I\setminus D\left(  \alpha\right)  \right\vert }\\
&  =%
\begin{cases}
2^{\left\vert D\left(  \beta\right)  \right\vert }, & \text{if }D\left(
\beta\right)  \subseteq D\left(  \alpha\right)  ;\\
0, & \text{otherwise}%
\end{cases}
\ \ \ \ \ \ \ \ \ \ \left(  \text{by Lemma \ref{lem.eta.through-F.lem1}%
}\right)  .
\end{align*}
Hence, this becomes%
\begin{align*}
&  2\sum_{\gamma\in\operatorname*{Comp}\nolimits_{n}}\left(  -1\right)
^{\left\vert D\left(  \gamma\right)  \setminus D\left(  \alpha\right)
\right\vert }L_{\gamma}\\
&  =2\sum_{\beta\in\operatorname*{Comp}\nolimits_{n}}\underbrace{\sum
_{\substack{\gamma\in\operatorname*{Comp}\nolimits_{n};\\D\left(
\beta\right)  \supseteq D\left(  \gamma\right)  }}\left(  -1\right)
^{\left\vert D\left(  \gamma\right)  \setminus D\left(  \alpha\right)
\right\vert }}_{=%
\begin{cases}
2^{\left\vert D\left(  \beta\right)  \right\vert }, & \text{if }D\left(
\beta\right)  \subseteq D\left(  \alpha\right)  ;\\
0, & \text{otherwise}%
\end{cases}
}M_{\beta}\\
&  =2\sum_{\beta\in\operatorname*{Comp}\nolimits_{n}}%
\begin{cases}
2^{\left\vert D\left(  \beta\right)  \right\vert }, & \text{if }D\left(
\beta\right)  \subseteq D\left(  \alpha\right)  ;\\
0, & \text{otherwise}%
\end{cases}
\ M_{\beta}\\
&  =2\sum_{\substack{\beta\in\operatorname*{Comp}\nolimits_{n};\\D\left(
\beta\right)  \subseteq D\left(  \alpha\right)  }}2^{\left\vert D\left(
\beta\right)  \right\vert }M_{\beta}=\sum_{\substack{\beta\in
\operatorname*{Comp}\nolimits_{n};\\D\left(  \beta\right)  \subseteq D\left(
\alpha\right)  }}\underbrace{2^{\left\vert D\left(  \beta\right)  \right\vert
+1}}_{\substack{=2^{\ell\left(  \beta\right)  }\\\text{(since }\left\vert
D\left(  \beta\right)  \right\vert +1=\ell\left(  \beta\right)  \text{)}%
}}M_{\beta}\\
&  =\sum_{\substack{\beta\in\operatorname*{Comp}\nolimits_{n};\\D\left(
\beta\right)  \subseteq D\left(  \alpha\right)  }}2^{\ell\left(  \beta\right)
}M_{\beta}=\eta_{\alpha}\ \ \ \ \ \ \ \ \ \ \left(  \text{by the definition of
}\eta_{\alpha}\right)  .
\end{align*}

\end{proof}

\subsection{The antipode of $\eta_{\alpha}$}

The \emph{antipode} of $\operatorname*{QSym}$ is a certain $\mathbf{k}$-linear
map $S:\operatorname*{QSym}\rightarrow\operatorname*{QSym}$ that can be
defined in terms of the Hopf algebra structure of $\operatorname*{QSym}$,
which we have not defined so far. But there are various formulas for its
values on certain quasisymmetric functions that can be used as alternative
definitions. For example, for any $n\in\mathbb{N}$ and any $\alpha=\left(
\alpha_{1},\alpha_{2},\ldots,\alpha_{\ell}\right)  \in\operatorname*{Comp}%
\nolimits_{n}$, we have%
\[
S\left(  M_{\alpha}\right)  =\left(  -1\right)  ^{\ell}\sum_{\substack{\gamma
\in\operatorname*{Comp}\nolimits_{n};\\D\left(  \gamma\right)  \subseteq
D\left(  \alpha_{\ell},\alpha_{\ell-1},\ldots,\alpha_{1}\right)  }}M_{\gamma
}.
\]
This can be used to define $S$ (since $S$ is to be $\mathbf{k}$-linear). Also,
for each composition $\alpha$, we have $S\left(  L_{\alpha}\right)  =\left(
-1\right)  ^{\left\vert \alpha\right\vert }L_{\omega\left(  \alpha\right)  }$,
where $\omega\left(  \alpha\right)  $ is a certain composition known as the
\emph{complement} of $\alpha$. See \cite[Theorem 5.1.11 and Proposition
5.2.15]{GriRei} for details and proofs. Note that $S$ is a $\mathbf{k}%
$-algebra homomorphism and an involution (that is, $S^{2}=\operatorname*{id}%
$). (Again, this is derived from abstract algebraic properties of antipodes in
\cite{GriRei}, but can also be showed more directly.)

\begin{definition}
If $\alpha=\left(  \alpha_{1},\alpha_{2},\ldots,\alpha_{\ell}\right)  $ is a
composition, then the \emph{reversal} of $\alpha$ is defined to be the
composition $\left(  \alpha_{\ell},\alpha_{\ell-1},\ldots,\alpha_{1}\right)
$. It is denoted by $\operatorname*{rev}\alpha$.
\end{definition}

\begin{proposition}
\label{prop.eta.S}Let $\alpha\in\operatorname*{Comp}$. Then, the antipode $S$
of $\operatorname*{QSym}$ satisfies
\[
S\left(  \eta_{\alpha}\right)  =\left(  -1\right)  ^{\ell\left(
\alpha\right)  }\eta_{\operatorname*{rev}\alpha}.
\]

\end{proposition}

\begin{proof}
TODO. (This follows easily from Proposition \ref{prop.eta.through-F}.)
\end{proof}

Proposition \ref{prop.eta.S} generalizes \cite[Proposition 2.9]{Hsiao07}.

\subsection{The $\eta_{\alpha}$ as a basis}

\begin{theorem}
\label{thm.eta.basis}Assume that $2$ is invertible in $\mathbf{k}$. Then, the
family $\left(  \eta_{\alpha}\right)  _{\alpha\in\operatorname*{Comp}}$ is a
basis of the $\mathbf{k}$-module $\operatorname*{QSym}$.
\end{theorem}

\begin{proof}
TODO. Fix $n\in\mathbb{N}$. Consider the $n$-th graded component
$\operatorname*{QSym}\nolimits_{n}$ of $\operatorname*{QSym}$.

Define a partial order $\prec$ on the finite set $\operatorname*{Comp}%
\nolimits_{n}$ by setting $\beta\prec\alpha$ if and only if $\ell\left(
\beta\right)  <\ell\left(  \alpha\right)  $

The definition of $\eta_{\alpha}$ shows that%
\begin{align*}
\eta_{\alpha}  &  =2^{\ell\left(  \alpha\right)  }M_{\alpha}+\left(  \text{a
linear combination of }M_{\beta}\text{ with }\beta\in\operatorname*{Comp}%
\nolimits_{n}\text{ satisfying }\ell\left(  \beta\right)  <\ell\left(
\alpha\right)  \right) \\
&  =2^{\ell\left(  \alpha\right)  }M_{\alpha}+\left(  \text{a linear
combination of }M_{\beta}\text{ with }\beta\in\operatorname*{Comp}%
\nolimits_{n}\text{ satisfying }\beta\prec\alpha\right)
\end{align*}
for each $\alpha\in\operatorname*{Comp}\nolimits_{n}$. Thus, the family
$\left(  \eta_{\alpha}\right)  _{\alpha\in\operatorname*{Comp}\nolimits_{n}}$
expands invertibly triangularly in the family $\left(  M_{\alpha}\right)
_{\alpha\in\operatorname*{Comp}\nolimits_{n}}$ with respect to the partial
order $\prec$ (where we are using the terminology from \cite[\S 11.1]%
{GriRei}). Hence, \cite[Corollary 11.1.19(e)]{GriRei} shows that the family
$\left(  \eta_{\alpha}\right)  _{\alpha\in\operatorname*{Comp}\nolimits_{n}}$
is a basis of the $\mathbf{k}$-module $\operatorname*{QSym}\nolimits_{n}$
(since the family $\left(  M_{\alpha}\right)  _{\alpha\in\operatorname*{Comp}%
\nolimits_{n}}$ is a basis of $\operatorname*{QSym}\nolimits_{n}$).

Forget that we fixed $n$. Thus, we have shown that the family $\left(
\eta_{\alpha}\right)  _{\alpha\in\operatorname*{Comp}\nolimits_{n}}$ is a
basis of the $\mathbf{k}$-module $\operatorname*{QSym}\nolimits_{n}$ for each
$n\in\mathbb{N}$. Hence, the family $\left(  \eta_{\alpha}\right)  _{\alpha
\in\operatorname*{Comp}}$ is a basis of the $\mathbf{k}$-module $\bigoplus
_{n\in\mathbb{N}}\operatorname*{QSym}\nolimits_{n}=\operatorname*{QSym}$. This
proves Theorem \ref{thm.eta.basis}.
\end{proof}

We can explicitly expand the monomial quasisymmetric functions $M_{\beta}$ in
the basis $\left(  \eta_{\alpha}\right)  _{\alpha\in\operatorname*{Comp}}$:

\begin{proposition}
\label{prop.eta.M-through-eta}Let $n\in\mathbb{N}$. Let $\beta\in
\operatorname*{Comp}\nolimits_{n}$ be a composition. Then,%
\[
2^{\ell\left(  \beta\right)  }M_{\beta}=\sum_{\substack{\alpha\in
\operatorname*{Comp}\nolimits_{n};\\D\left(  \alpha\right)  \subseteq D\left(
\beta\right)  }}\left(  -1\right)  ^{\ell\left(  \beta\right)  -\ell\left(
\alpha\right)  }\eta_{\alpha}.
\]

\end{proposition}

\begin{proof}
TODO. (Follows from (\ref{eq.def.etaalpha.def}) using M\"{o}bius inversion.)
\end{proof}

\subsection{The product rule}

Next comes a fairly nontrivial result: Given two compositions $\alpha$ and
$\beta$, the product $\eta_{\alpha}\eta_{\beta}$ is a $\mathbf{k}$-linear
combination of the family $\left(  \eta_{\gamma}\right)  _{\gamma
\in\operatorname*{Comp}}$. If $2$ is invertible in $\mathbf{k}$, this follows
from Theorem \ref{thm.eta.basis}, but in the general case (thus, e.g., for
$\mathbf{k}=\mathbb{Z}$), I don't see any simple reasons why this should hold.
Nevertheless it does, and there is a combinatorial expression. To state it, we
need a weird variant of shuffles that I have never seen in the literature.
First, as inspiration, let me cite the analogous rule for products of the form
$M_{\alpha}M_{\beta}$:

\begin{definition}
\label{def.stuffler}Let $\alpha=\left(  \alpha_{1},\alpha_{2},\ldots
,\alpha_{\ell}\right)  $ and $\beta=\left(  \beta_{1},\beta_{2},\ldots
,\beta_{m}\right)  $ be two compositions.

Fix two chains (i.e., totally ordered sets) $\left\{  p_{1}<p_{2}%
<\cdots<p_{\ell}\right\}  $ and $\left\{  q_{1}<q_{2}<\cdots<q_{m}\right\}  $,
and let%
\[
D=\left\{  p_{1}<p_{2}<\cdots<p_{\ell}\right\}  \sqcup\left\{  q_{1}%
<q_{2}<\cdots<q_{m}\right\}
\]
be their disjoint union. This $D$ is a poset with $\ell+m$ elements
$p_{1},p_{2},\ldots,p_{\ell},q_{1},q_{2},\ldots,q_{m}$, whose relations are
given by $p_{1}<p_{2}<\cdots<p_{\ell}$ and $q_{1}<q_{2}<\cdots<q_{m}$ (while
each $p_{i}$ is incomparable to each $q_{j}$).

A \emph{stuffler} for $\alpha$ and $\beta$ shall mean a surjective and
strictly order-preserving map%
\[
f:D\rightarrow\left\{  1<2<\cdots<k\right\}  \ \ \ \ \ \ \ \ \ \ \text{for
some }k\in\mathbb{N}.
\]
(\textquotedblleft Strictly order-preserving\textquotedblright\ means that if
$u$ and $v$ are two elements of the poset $D$ satisfying $u<v$, then $f\left(
u\right)  <f\left(  v\right)  $.)

If $f:D\rightarrow\left\{  1<2<\cdots<k\right\}  $ is a stuffler for $\alpha$
and $\beta$, then we define the \emph{weight} $\operatorname*{wt}\left(
f\right)  $ of the stuffler $f$ to be the composition $\left(
\operatorname*{wt}\nolimits_{1}\left(  f\right)  ,\operatorname*{wt}%
\nolimits_{2}\left(  f\right)  ,\ldots,\operatorname*{wt}\nolimits_{k}\left(
f\right)  \right)  $, where%
\[
\operatorname*{wt}\nolimits_{s}\left(  f\right)  =\sum_{\substack{u\in\left[
\ell\right]  ;\\f\left(  p_{u}\right)  =s}}\alpha_{u}+\sum_{\substack{v\in
\left[  m\right]  ;\\f\left(  q_{v}\right)  =s}}\beta_{v}%
\ \ \ \ \ \ \ \ \ \ \text{for each }s\in\left[  k\right]  .
\]
Note that each of the two sums on the right hand side has at most $1$ addend.
\end{definition}

\begin{example}
Let $\alpha=\left(  4,2\right)  $ and $\beta=\left(  1,3,1\right)  $ be two
compositions. Then, the poset $D$ in Definition \ref{def.stuffler} is
$D=\left\{  p_{1}<p_{2}\right\}  \sqcup\left\{  q_{1}<q_{2}<q_{3}\right\}  $.
The following maps (written in two-line notation) are stufflers for $\alpha$
and $\beta$:%
\begin{align*}
&  \left(
\begin{array}
[c]{ccccc}%
p_{1} & p_{2} & q_{1} & q_{2} & q_{3}\\
1 & 2 & 3 & 4 & 5
\end{array}
\right)  ,\ \ \ \ \ \ \ \ \ \ \left(
\begin{array}
[c]{ccccc}%
p_{1} & p_{2} & q_{1} & q_{2} & q_{3}\\
1 & 4 & 2 & 3 & 5
\end{array}
\right)  ,\\
&  \left(
\begin{array}
[c]{ccccc}%
p_{1} & p_{2} & q_{1} & q_{2} & q_{3}\\
2 & 5 & 1 & 3 & 4
\end{array}
\right)  ,\ \ \ \ \ \ \ \ \ \ \left(
\begin{array}
[c]{ccccc}%
p_{1} & p_{2} & q_{1} & q_{2} & q_{3}\\
1 & 2 & 1 & 3 & 4
\end{array}
\right)  ,\\
&  \left(
\begin{array}
[c]{ccccc}%
p_{1} & p_{2} & q_{1} & q_{2} & q_{3}\\
1 & 3 & 2 & 3 & 4
\end{array}
\right)  ,\ \ \ \ \ \ \ \ \ \ \left(
\begin{array}
[c]{ccccc}%
p_{1} & p_{2} & q_{1} & q_{2} & q_{3}\\
1 & 3 & 1 & 2 & 3
\end{array}
\right)  .
\end{align*}
(The list is not exhaustive -- there are many more stufflers for $\alpha$ and
$\beta$.) On the other hand, $\left(
\begin{array}
[c]{ccccc}%
p_{1} & p_{2} & q_{1} & q_{2} & q_{3}\\
1 & 4 & 2 & 2 & 3
\end{array}
\right)  $ is not a stuffler for $\alpha$ and $\beta$ (since it fails the
\textquotedblleft strictly order-preserving\textquotedblright\ condition, by
way of sending $q_{1}$ and $q_{2}$ to the same number), and $\left(
\begin{array}
[c]{ccccc}%
p_{1} & p_{2} & q_{1} & q_{2} & q_{3}\\
2 & 4 & 2 & 3 & 4
\end{array}
\right)  $ is not a stuffler either (since it fails to be surjective onto
$\left\{  1<2<3<4\right\}  $).

Here are the weights of the above listed stufflers:%
\begin{align*}
\operatorname*{wt}\left(
\begin{array}
[c]{ccccc}%
p_{1} & p_{2} & q_{1} & q_{2} & q_{3}\\
1 & 2 & 3 & 4 & 5
\end{array}
\right)   &  =\left(  4,2,1,3,1\right)  ,\\
\operatorname*{wt}\left(
\begin{array}
[c]{ccccc}%
p_{1} & p_{2} & q_{1} & q_{2} & q_{3}\\
1 & 4 & 2 & 3 & 5
\end{array}
\right)   &  =\left(  4,1,3,2,1\right)  ,\\
\operatorname*{wt}\left(
\begin{array}
[c]{ccccc}%
p_{1} & p_{2} & q_{1} & q_{2} & q_{3}\\
2 & 5 & 1 & 3 & 4
\end{array}
\right)   &  =\left(  1,4,3,1,2\right)  ,\\
\operatorname*{wt}\left(
\begin{array}
[c]{ccccc}%
p_{1} & p_{2} & q_{1} & q_{2} & q_{3}\\
1 & 2 & 1 & 3 & 4
\end{array}
\right)   &  =\left(  4+1,2,3,1\right)  =\left(  5,2,3,1\right)  ,\\
\operatorname*{wt}\left(
\begin{array}
[c]{ccccc}%
p_{1} & p_{2} & q_{1} & q_{2} & q_{3}\\
1 & 3 & 2 & 3 & 4
\end{array}
\right)   &  =\left(  4,1,2+3,1\right)  =\left(  4,1,5,1\right)  ,\\
\operatorname*{wt}\left(
\begin{array}
[c]{ccccc}%
p_{1} & p_{2} & q_{1} & q_{2} & q_{3}\\
1 & 3 & 1 & 2 & 3
\end{array}
\right)   &  =\left(  4+1,3,2+1\right)  =\left(  5,3,3\right)  .
\end{align*}

\end{example}

The composition $\operatorname*{wt}\left(  f\right)  $ in Definition
\ref{def.stuffler} is called a \emph{stuffle} (or \emph{overlapping shuffle})
of $\alpha$ with $\beta$. Each of its entries is either an entry of $\alpha$
or an entry of $\beta$ or a sum of an entry of $\alpha$ with an entry of
$\beta$; moreover, each of the entries of $\alpha$ and of $\beta$ is used in
exactly one entry of $\operatorname*{wt}\left(  f\right)  $, and the entries
of $\alpha$ appear in their original order in the entries of
$\operatorname*{wt}\left(  f\right)  $, and so do the entries of $\beta$.

Now we can state the multiplication rule for products of the form $M_{\alpha
}M_{\beta}$ (\cite[Proposition 5.1.3]{GriRei}):

\begin{theorem}
\label{thm.MaMb}Let $\alpha$ and $\beta$ be two compositions. Then,%
\[
M_{\alpha}M_{\beta}=\sum_{f\text{ is a stuffler for }\alpha\text{ and }\beta
}M_{\operatorname*{wt}\left(  f\right)  }.
\]

\end{theorem}

\begin{example}
Let $\alpha=\left(  a,b\right)  $ and $\beta=\left(  c,d\right)  $ be two
compositions of length $2$. Let us compute $M_{\left(  a,b\right)  }M_{\left(
c,d\right)  }$ using Theorem \ref{thm.MaMb}. The stufflers for $\alpha$ and
$\beta$ are the maps (written here in two-line notation)%
\begin{align*}
&  \left(
\begin{array}
[c]{cccc}%
p_{1} & p_{2} & q_{1} & q_{2}\\
1 & 2 & 3 & 4
\end{array}
\right)  ,\ \ \ \ \ \ \ \ \ \ \left(
\begin{array}
[c]{cccc}%
p_{1} & p_{2} & q_{1} & q_{2}\\
1 & 3 & 2 & 4
\end{array}
\right)  ,\ \ \ \ \ \ \ \ \ \ \left(
\begin{array}
[c]{cccc}%
p_{1} & p_{2} & q_{1} & q_{2}\\
1 & 4 & 2 & 3
\end{array}
\right)  ,\\
&  \left(
\begin{array}
[c]{cccc}%
p_{1} & p_{2} & q_{1} & q_{2}\\
2 & 3 & 1 & 4
\end{array}
\right)  ,\ \ \ \ \ \ \ \ \ \ \left(
\begin{array}
[c]{cccc}%
p_{1} & p_{2} & q_{1} & q_{2}\\
2 & 4 & 1 & 3
\end{array}
\right)  ,\ \ \ \ \ \ \ \ \ \ \left(
\begin{array}
[c]{cccc}%
p_{1} & p_{2} & q_{1} & q_{2}\\
3 & 4 & 1 & 2
\end{array}
\right)  ,\\
&  \left(
\begin{array}
[c]{cccc}%
p_{1} & p_{2} & q_{1} & q_{2}\\
1 & 2 & 1 & 3
\end{array}
\right)  ,\ \ \ \ \ \ \ \ \ \ \left(
\begin{array}
[c]{cccc}%
p_{1} & p_{2} & q_{1} & q_{2}\\
1 & 3 & 1 & 2
\end{array}
\right)  ,\ \ \ \ \ \ \ \ \ \ \left(
\begin{array}
[c]{cccc}%
p_{1} & p_{2} & q_{1} & q_{2}\\
1 & 2 & 2 & 3
\end{array}
\right)  ,\\
&  \left(
\begin{array}
[c]{cccc}%
p_{1} & p_{2} & q_{1} & q_{2}\\
2 & 3 & 1 & 2
\end{array}
\right)  ,\ \ \ \ \ \ \ \ \ \ \left(
\begin{array}
[c]{cccc}%
p_{1} & p_{2} & q_{1} & q_{2}\\
1 & 3 & 2 & 3
\end{array}
\right)  ,\ \ \ \ \ \ \ \ \ \ \left(
\begin{array}
[c]{cccc}%
p_{1} & p_{2} & q_{1} & q_{2}\\
2 & 3 & 1 & 3
\end{array}
\right)  ,\\
&  \left(
\begin{array}
[c]{cccc}%
p_{1} & p_{2} & q_{1} & q_{2}\\
1 & 2 & 1 & 2
\end{array}
\right)  .
\end{align*}
Their respective weights are
\begin{align*}
&  \left(  a,b,c,d\right)  ,\ \ \ \ \ \ \ \ \ \ \left(  a,c,b,d\right)
,\ \ \ \ \ \ \ \ \ \ \left(  a,c,d,b\right)  ,\ \ \ \ \ \ \ \ \ \ \left(
c,a,b,d\right)  ,\\
&  \left(  c,a,d,b\right)  ,\ \ \ \ \ \ \ \ \ \ \left(  c,d,a,b\right)
,\ \ \ \ \ \ \ \ \ \ \left(  a+c,b,d\right)  ,\ \ \ \ \ \ \ \ \ \ \left(
a+c,d,b\right)  ,\\
&  \left(  a,b+c,d\right)  ,\ \ \ \ \ \ \ \ \ \ \left(  c,a+d,b\right)
,\ \ \ \ \ \ \ \ \ \ \left(  a,c,b+d\right)  ,\ \ \ \ \ \ \ \ \ \ \left(
c,a,b+d\right)  ,\\
&  \left(  a+c,b+d\right)  .
\end{align*}
Thus, Theorem \ref{thm.MaMb} yields%
\begin{align*}
M_{\left(  a,b\right)  }M_{\left(  c,d\right)  }  &  =M_{\left(
a,b,c,d\right)  }+M_{\left(  a,c,b,d\right)  }+M_{\left(  a,c,d,b\right)
}+M_{\left(  c,a,b,d\right)  }+M_{\left(  c,a,d,b\right)  }+M_{\left(
c,d,a,b\right)  }\\
&  \ \ \ \ \ \ \ \ \ \ +M_{\left(  a+c,b,d\right)  }+M_{\left(
a+c,d,b\right)  }+M_{\left(  a,b+c,d\right)  }+M_{\left(  c,a+d,b\right)  }\\
&  \ \ \ \ \ \ \ \ \ \ +M_{\left(  a,c,b+d\right)  }+M_{\left(
c,a,b+d\right)  }+M_{\left(  a+c,b+d\right)  }.
\end{align*}

\end{example}

The formula for $\eta_{\alpha}\eta_{\beta}$ is similar but subtler. Instead of
stufflers, we need what I call the \emph{liminal stufflers}, which are defined
as follows:

\begin{definition}
\label{def.limstuffler}Let $\alpha=\left(  \alpha_{1},\alpha_{2},\ldots
,\alpha_{\ell}\right)  $ and $\beta=\left(  \beta_{1},\beta_{2},\ldots
,\beta_{m}\right)  $ be two compositions.

Fix two chains (i.e., totally ordered sets) $\left\{  p_{1}<p_{2}%
<\cdots<p_{\ell}\right\}  $ and $\left\{  q_{1}<q_{2}<\cdots<q_{m}\right\}  $,
and define a poset $D$ as in Definition \ref{def.stuffler}.

A \emph{liminal stuffler} for $\alpha$ and $\beta$ shall mean a surjective and
weakly order-preserving map%
\[
f:D\rightarrow\left\{  1<2<\cdots<k\right\}  \ \ \ \ \ \ \ \ \ \ \text{for
some }k\in\mathbb{N}%
\]
with the property that each $s\in\left\{  1<2<\cdots<k\right\}  $ satisfies%
\begin{align}
&  \left\vert \left\{  u\in\left[  \ell\right]  \ \mid\ f\left(  p_{u}\right)
=s\right\}  \right\vert -\left\vert \left\{  v\in\left[  m\right]
\ \mid\ f\left(  q_{v}\right)  =s\right\}  \right\vert \nonumber\\
&  \in\left\{  1,-1\right\}  . \label{eq.def.limstuffler.liminality}%
\end{align}


(\textquotedblleft Weakly order-preserving\textquotedblright\ means that if
$u$ and $v$ are two elements of the poset $D$ satisfying $u<v$, then $f\left(
u\right)  \leq f\left(  v\right)  $.)

If $f:D\rightarrow\left\{  1<2<\cdots<k\right\}  $ is a liminal stuffler for
$\alpha$ and $\beta$, then we define the \emph{weight} $\operatorname*{wt}%
\left(  f\right)  $ of $f$ to be the composition $\left(  \operatorname*{wt}%
\nolimits_{1}\left(  f\right)  ,\operatorname*{wt}\nolimits_{2}\left(
f\right)  ,\ldots,\operatorname*{wt}\nolimits_{k}\left(  f\right)  \right)  $,
where%
\[
\operatorname*{wt}\nolimits_{s}\left(  f\right)  =\sum_{\substack{u\in\left[
\ell\right]  ;\\f\left(  p_{u}\right)  =s}}\alpha_{u}+\sum_{\substack{v\in
\left[  m\right]  ;\\f\left(  q_{v}\right)  =s}}\beta_{v}%
\ \ \ \ \ \ \ \ \ \ \text{for each }s\in\left[  k\right]  .
\]
(This time, the sums on the right hand side can have more than $1$ addend. But
(\ref{eq.def.limstuffler.liminality}) ensures that one of the two sums has
exactly $1$ more addend than the other.)

If $f:D\rightarrow\left\{  1<2<\cdots<k\right\}  $ is a liminal stuffler for
$\alpha$ and $\beta$, then the \emph{loss} of $f$ is defined to be the
nonnegative integer%
\[
\sum_{s=1}^{k}\min\left\{  \left\vert \left\{  u\in\left[  \ell\right]
\ \mid\ f\left(  p_{u}\right)  =s\right\}  \right\vert ,\ \ \left\vert
\left\{  v\in\left[  m\right]  \ \mid\ f\left(  q_{v}\right)  =s\right\}
\right\vert \right\}  .
\]
This is denoted by $\operatorname*{loss}\left(  f\right)  $. It is easy to see
that $k=\ell+m-2\operatorname*{loss}\left(  f\right)  $.
\end{definition}

\begin{example}
Let $\alpha=\left(  4,2\right)  $ and $\beta=\left(  1,3,1\right)  $ be two
compositions. Then, the poset $D$ in Definition \ref{def.stuffler} is
$D=\left\{  p_{1}<p_{2}\right\}  \sqcup\left\{  q_{1}<q_{2}<q_{3}\right\}  $.
The following maps (written in two-line notation) are liminal stufflers for
$\alpha$ and $\beta$:%
\begin{align*}
&  \left(
\begin{array}
[c]{ccccc}%
p_{1} & p_{2} & q_{1} & q_{2} & q_{3}\\
1 & 2 & 3 & 4 & 5
\end{array}
\right)  ,\ \ \ \ \ \ \ \ \ \ \left(
\begin{array}
[c]{ccccc}%
p_{1} & p_{2} & q_{1} & q_{2} & q_{3}\\
2 & 5 & 1 & 3 & 4
\end{array}
\right)  ,\\
&  \left(
\begin{array}
[c]{ccccc}%
p_{1} & p_{2} & q_{1} & q_{2} & q_{3}\\
1 & 1 & 1 & 2 & 3
\end{array}
\right)  ,\ \ \ \ \ \ \ \ \ \ \left(
\begin{array}
[c]{ccccc}%
p_{1} & p_{2} & q_{1} & q_{2} & q_{3}\\
1 & 2 & 2 & 2 & 3
\end{array}
\right)  ,\\
&  \left(
\begin{array}
[c]{ccccc}%
p_{1} & p_{2} & q_{1} & q_{2} & q_{3}\\
2 & 2 & 1 & 2 & 3
\end{array}
\right)  ,\ \ \ \ \ \ \ \ \ \ \left(
\begin{array}
[c]{ccccc}%
p_{1} & p_{2} & q_{1} & q_{2} & q_{3}\\
1 & 1 & 1 & 1 & 1
\end{array}
\right)  .
\end{align*}
(The list is not exhaustive -- there are many more liminal stufflers for
$\alpha$ and $\beta$. In particular, any \textbf{injective} stuffler for
$\alpha$ and $\beta$ is a liminal stuffler for $\alpha$ and $\beta$ as well.)
On the other hand, $\left(
\begin{array}
[c]{ccccc}%
p_{1} & p_{2} & q_{1} & q_{2} & q_{3}\\
1 & 1 & 1 & 1 & 2
\end{array}
\right)  $ is not a liminal stuffler for $\alpha$ and $\beta$ (since
(\ref{eq.def.limstuffler.liminality}) fails for $s=1$), and $\left(
\begin{array}
[c]{ccccc}%
p_{1} & p_{2} & q_{1} & q_{2} & q_{3}\\
2 & 2 & 2 & 2 & 2
\end{array}
\right)  $ is not a liminal stuffler either (since it fails to be surjective
onto $\left\{  1<2<3<4\right\}  $).

Here are the weights of the above listed liminal stufflers:%
\begin{align*}
\operatorname*{wt}\left(
\begin{array}
[c]{ccccc}%
p_{1} & p_{2} & q_{1} & q_{2} & q_{3}\\
1 & 2 & 3 & 4 & 5
\end{array}
\right)   &  =\left(  4,2,1,3,1\right)  ,\\
\operatorname*{wt}\left(
\begin{array}
[c]{ccccc}%
p_{1} & p_{2} & q_{1} & q_{2} & q_{3}\\
2 & 5 & 1 & 3 & 4
\end{array}
\right)   &  =\left(  1,4,3,1,2\right)  ,\\
\operatorname*{wt}\left(
\begin{array}
[c]{ccccc}%
p_{1} & p_{2} & q_{1} & q_{2} & q_{3}\\
1 & 1 & 1 & 2 & 3
\end{array}
\right)   &  =\left(  4+2+1,3,1\right)  =\left(  7,3,1\right)  ,\\
\operatorname*{wt}\left(
\begin{array}
[c]{ccccc}%
p_{1} & p_{2} & q_{1} & q_{2} & q_{3}\\
1 & 2 & 2 & 2 & 3
\end{array}
\right)   &  =\left(  4,2+1+3,1\right)  =\left(  4,6,1\right)  ,\\
\operatorname*{wt}\left(
\begin{array}
[c]{ccccc}%
p_{1} & p_{2} & q_{1} & q_{2} & q_{3}\\
2 & 2 & 1 & 2 & 3
\end{array}
\right)   &  =\left(  1,4+2+3,1\right)  =\left(  1,9,1\right)  ,\\
\operatorname*{wt}\left(
\begin{array}
[c]{ccccc}%
p_{1} & p_{2} & q_{1} & q_{2} & q_{3}\\
1 & 1 & 1 & 1 & 1
\end{array}
\right)   &  =\left(  4+2+1+3+1\right)  =\left(  11\right)  .
\end{align*}
The losses of these liminal stufflers are $0$, $0$, $1$, $1$, $1$ and $2$, respectively.
\end{example}

Intuitively, the composition $\operatorname*{wt}\left(  f\right)  $ in
Definition \ref{def.limstuffler} can be thought of as a variant of a stuffle
of $\alpha$ with $\beta$, but instead of adding an entry of $\alpha$ with an
entry of $\beta$, it allows adding $i$ consecutive entries of $\alpha$ and $j$
consecutive entries of $\beta$ whenever $i$ and $j$ are integers satisfying
$i-j\in\left\{  1,-1\right\}  $. The statistic $\operatorname*{loss}\left(
f\right)  $ tells how much is being added, i.e., how far this
\textquotedblleft stuffle\textquotedblright\ deviates from a shuffle.

Now we can state the multiplication rule for products of the form
$\eta_{\alpha}\eta_{\beta}$:

\begin{theorem}
\label{thm.etaaetab}Let $\alpha$ and $\beta$ be two compositions. Then,%
\[
\eta_{\alpha}\eta_{\beta}=\sum_{\substack{f\text{ is a liminal stuffler}%
\\\text{for }\alpha\text{ and }\beta}}\left(  -1\right)
^{\operatorname*{loss}\left(  f\right)  }\eta_{\operatorname*{wt}\left(
f\right)  }.
\]

\end{theorem}

\begin{example}
Let $\alpha=\left(  a,b\right)  $ and $\beta=\left(  c,d\right)  $ be two
compositions of length $2$. Let us compute $\eta_{\left(  a,b\right)  }%
\eta_{\left(  c,d\right)  }$ using Theorem \ref{thm.etaaetab}. The liminal
stufflers for $\alpha$ and $\beta$ are the maps (written here in two-line
notation)%
\begin{align*}
&  \left(
\begin{array}
[c]{cccc}%
p_{1} & p_{2} & q_{1} & q_{2}\\
1 & 2 & 3 & 4
\end{array}
\right)  ,\ \ \ \ \ \ \ \ \ \ \left(
\begin{array}
[c]{cccc}%
p_{1} & p_{2} & q_{1} & q_{2}\\
1 & 3 & 2 & 4
\end{array}
\right)  ,\ \ \ \ \ \ \ \ \ \ \left(
\begin{array}
[c]{cccc}%
p_{1} & p_{2} & q_{1} & q_{2}\\
1 & 4 & 2 & 3
\end{array}
\right)  ,\\
&  \left(
\begin{array}
[c]{cccc}%
p_{1} & p_{2} & q_{1} & q_{2}\\
2 & 3 & 1 & 4
\end{array}
\right)  ,\ \ \ \ \ \ \ \ \ \ \left(
\begin{array}
[c]{cccc}%
p_{1} & p_{2} & q_{1} & q_{2}\\
2 & 4 & 1 & 3
\end{array}
\right)  ,\ \ \ \ \ \ \ \ \ \ \left(
\begin{array}
[c]{cccc}%
p_{1} & p_{2} & q_{1} & q_{2}\\
3 & 4 & 1 & 2
\end{array}
\right)  ,\\
&  \left(
\begin{array}
[c]{cccc}%
p_{1} & p_{2} & q_{1} & q_{2}\\
1 & 2 & 2 & 2
\end{array}
\right)  ,\ \ \ \ \ \ \ \ \ \ \left(
\begin{array}
[c]{cccc}%
p_{1} & p_{2} & q_{1} & q_{2}\\
2 & 2 & 1 & 2
\end{array}
\right)  ,\\
&  \left(
\begin{array}
[c]{cccc}%
p_{1} & p_{2} & q_{1} & q_{2}\\
1 & 1 & 1 & 2
\end{array}
\right)  ,\ \ \ \ \ \ \ \ \ \ \left(
\begin{array}
[c]{cccc}%
p_{1} & p_{2} & q_{1} & q_{2}\\
1 & 2 & 1 & 1
\end{array}
\right)  .
\end{align*}
Their respective weights are
\begin{align*}
&  \left(  a,b,c,d\right)  ,\ \ \ \ \ \ \ \ \ \ \left(  a,c,b,d\right)
,\ \ \ \ \ \ \ \ \ \ \left(  a,c,d,b\right)  ,\ \ \ \ \ \ \ \ \ \ \left(
c,a,b,d\right)  ,\\
&  \left(  c,a,d,b\right)  ,\ \ \ \ \ \ \ \ \ \ \left(  c,d,a,b\right)
,\ \ \ \ \ \ \ \ \ \ \left(  a,b+c+d\right)  ,\\
&  \left(  c,a+b+d\right)  ,\ \ \ \ \ \ \ \ \ \ \left(  a+b+c,d\right)
,\ \ \ \ \ \ \ \ \ \ \left(  a+c+d,b\right)  ,
\end{align*}
and their respective losses are $0,0,0,0,0,0,1,1,1,1$. Thus, Theorem
\ref{thm.etaaetab} yields%
\begin{align*}
\eta_{\left(  a,b\right)  }\eta_{\left(  c,d\right)  }  &  =\eta_{\left(
a,b,c,d\right)  }+\eta_{\left(  a,c,b,d\right)  }+\eta_{\left(
a,c,d,b\right)  }+\eta_{\left(  c,a,b,d\right)  }+\eta_{\left(
c,a,d,b\right)  }+\eta_{\left(  c,d,a,b\right)  }\\
&  \ \ \ \ \ \ \ \ \ \ -\eta_{\left(  a,b+c+d\right)  }-\eta_{\left(
c,a+b+d\right)  }-\eta_{\left(  a+b+c,d\right)  }-\eta_{\left(
a+c+d,b\right)  }.
\end{align*}

\end{example}

We will prove Theorem \ref{thm.etaaetab} further below.

\begin{question}
Is there a direct/combinatorial proof of Theorem \ref{thm.etaaetab}, possibly
using Proposition \ref{prop.eta.through-x2} and a sign-reversing involution?
\end{question}

Theorem \ref{thm.etaaetab} can be seen to generalize \cite[Corollary
2.5]{Hsiao07}.

\subsection{The dual basis $\eta_{\alpha}^{\ast}$ in $\operatorname*{NSym}$}

In order to prove Theorem \ref{thm.etaaetab}, it suffices to prove it when
$\mathbf{k}=\mathbb{Q}$ (because all identities that hold in
$\operatorname*{QSym}$ over $\mathbb{Q}$ but involve no denominators will
automatically hold in $\operatorname*{QSym}$ over $\mathbb{Z}$, and therefore
also in $\operatorname*{QSym}$ over any commutative ring $\mathbf{k}$). Thus,
the following convention is harmless:

\begin{convention}
For the rest of this section, we WLOG assume that $2$ is invertible in
$\mathbf{k}$.
\end{convention}

Recall the Hopf algebra $\operatorname*{NSym}$ defined in \cite[\S 5.4]%
{GriRei}. It is the graded dual of the Hopf algebra $\operatorname*{QSym}$.

We will use only one basis of $\operatorname*{NSym}$, namely the basis
$\left(  H_{\alpha}\right)  _{\alpha\in\operatorname*{Comp}}$. This is the
basis of $\operatorname*{NSym}$ dual to the basis $\left(  M_{\alpha}\right)
_{\alpha\in\operatorname*{Comp}}$ of $\operatorname*{NSym}$. We write $H_{n}$
for $H_{\left(  n\right)  }$ whenever $n$ is a positive integer. We also set
$H_{0}=1$ and $H_{n}=0$ for all $n<0$.

(What I call $H_{\beta}$ is called $S_{\beta}$ in \cite{GKLLRT94}.)

\begin{definition}
For each $n\in\mathbb{N}$ and each composition $\alpha$ of $n$, we define
\[
\eta_{\alpha}^{\ast}=\sum_{\substack{\beta\in\operatorname*{Comp}%
\nolimits_{n};\\D\left(  \alpha\right)  \subseteq D\left(  \beta\right)
}}\dfrac{1}{2^{\ell\left(  \beta\right)  }}\left(  -1\right)  ^{\ell\left(
\beta\right)  -\ell\left(  \alpha\right)  }H_{\beta}\in\operatorname*{NSym}.
\]

\end{definition}

Then, it is straightforward to see the following:

\begin{proposition}
\label{prop.etastar-dual-basis}The family $\left(  \eta_{\alpha}^{\ast
}\right)  _{\alpha\in\operatorname*{Comp}}$ is the basis of
$\operatorname*{NSym}$ dual to the basis $\left(  \eta_{\alpha}\right)
_{\alpha\in\operatorname*{Comp}}$ of $\operatorname*{QSym}$.
\end{proposition}

\begin{proof}
TODO. (Follows from Proposition \ref{prop.eta.M-through-eta} by dualization.)
\end{proof}

\begin{definition}
For each positive integer $n$, let%
\[
\eta_{n}^{\ast}=\eta_{\left(  n\right)  }^{\ast}=\sum_{\beta\in
\operatorname*{Comp}\nolimits_{n}}\dfrac{1}{2^{\ell\left(  \beta\right)  }%
}\left(  -1\right)  ^{\ell\left(  \beta\right)  -1}H_{\beta}\in
\operatorname*{NSym}.
\]

\end{definition}

It turns out that we can easily express $\eta_{\alpha}^{\ast}$ for any
composition $\alpha$ using these $\eta_{n}^{\ast}$:

\begin{proposition}
\label{prop.etastar.mult}We have%
\[
\eta_{\alpha}^{\ast}=\eta_{\alpha_{1}}^{\ast}\eta_{\alpha_{2}}^{\ast}%
\cdots\eta_{\alpha_{k}}^{\ast}\ \ \ \ \ \ \ \ \ \ \text{for each composition
}\alpha=\left(  \alpha_{1},\alpha_{2},\ldots,\alpha_{k}\right)  .
\]

\end{proposition}

The main idea of the proof of Proposition \ref{prop.etastar.mult} is to
recognize that if $n=\left\vert \alpha\right\vert $, then the compositions
$\beta\in\operatorname*{Comp}\nolimits_{n}$ satisfying $D\left(
\alpha\right)  \subseteq D\left(  \beta\right)  $ are precisely the
compositions obtained from $\alpha$ by breaking up each entry of $\alpha$ into
pieces. A slicker way to formalize this proof proceeds using the notion of concatenation:

\begin{definition}
If $\alpha=\left(  \alpha_{1},\alpha_{2},\ldots,\alpha_{\ell}\right)  $ and
$\beta=\left(  \beta_{1},\beta_{2},\ldots,\beta_{k}\right)  $ are two
compositions, then the composition $\alpha\beta$ is defined by%
\[
\alpha\beta=\left(  \alpha_{1},\alpha_{2},\ldots,\alpha_{\ell},\beta_{1}%
,\beta_{2},\ldots,\beta_{k}\right)  .
\]
This composition $\alpha\beta$ is called the \emph{concatenation} of $\alpha$
and $\beta$. The operation of concatenation (sending any two compositions
$\alpha$ and $\beta$ to $\alpha\beta$) is associative, and the empty
composition $\varnothing$ is a neutral element for it; thus, the set of all
compositions is an abelian monoid under this operation.
\end{definition}

The following proposition is saying (in the jargon of combinatorial Hopf
algebras) that the basis $\left(  \eta_{\alpha}^{\ast}\right)  _{\alpha
\in\operatorname*{Comp}}$ of $\operatorname*{NSym}$ is multiplicative:

\begin{proposition}
\label{prop.etastar.concat2}Let $\alpha$ and $\beta$ be two compositions.
Then,%
\[
\eta_{\alpha}^{\ast}\eta_{\beta}^{\ast}=\eta_{\alpha\beta}^{\ast}.
\]

\end{proposition}

\begin{proof}
[Proof of Proposition \ref{prop.etastar.concat2}.]TODO. Sketch: The definition
of a concatenation easily yields that $\ell\left(  \gamma\right)  +\ell\left(
\delta\right)  =\ell\left(  \gamma\delta\right)  $ for any two compositions
$\gamma$ and $\delta$. Thus, in particular, $\ell\left(  \alpha\beta\right)
=\ell\left(  \alpha\right)  +\ell\left(  \beta\right)  $.

Let $n=\left\vert \alpha\right\vert $ and $m=\left\vert \beta\right\vert $.
Thus, $\alpha\in\operatorname*{Comp}\nolimits_{n}$ and $\beta\in
\operatorname*{Comp}\nolimits_{n}$, so that $\alpha\beta\in
\operatorname*{Comp}\nolimits_{n+m}$.

The definitions of $\eta_{\alpha}^{\ast}$ and $\eta_{\beta}^{\ast}$ yield%
\begin{align*}
\eta_{\alpha}^{\ast}  &  =\sum_{\substack{\gamma\in\operatorname*{Comp}%
\nolimits_{n};\\D\left(  \alpha\right)  \subseteq D\left(  \gamma\right)
}}\dfrac{1}{2^{\ell\left(  \gamma\right)  }}\left(  -1\right)  ^{\ell\left(
\gamma\right)  -\ell\left(  \alpha\right)  }H_{\gamma}%
\ \ \ \ \ \ \ \ \ \ \text{and}\\
\eta_{\beta}^{\ast}  &  =\sum_{\substack{\delta\in\operatorname*{Comp}%
\nolimits_{m};\\D\left(  \beta\right)  \subseteq D\left(  \delta\right)
}}\dfrac{1}{2^{\ell\left(  \delta\right)  }}\left(  -1\right)  ^{\ell\left(
\delta\right)  -\ell\left(  \beta\right)  }H_{\delta}.
\end{align*}
Multiplying these two equalities, we obtain%
\begin{align}
\eta_{\alpha}^{\ast}\eta_{\beta}^{\ast}  &  =\left(  \sum_{\substack{\gamma
\in\operatorname*{Comp}\nolimits_{n};\\D\left(  \alpha\right)  \subseteq
D\left(  \gamma\right)  }}\dfrac{1}{2^{\ell\left(  \gamma\right)  }}\left(
-1\right)  ^{\ell\left(  \gamma\right)  -\ell\left(  \alpha\right)  }%
H_{\gamma}\right)  \left(  \sum_{\substack{\delta\in\operatorname*{Comp}%
\nolimits_{m};\\D\left(  \beta\right)  \subseteq D\left(  \delta\right)
}}\dfrac{1}{2^{\ell\left(  \delta\right)  }}\left(  -1\right)  ^{\ell\left(
\delta\right)  -\ell\left(  \beta\right)  }H_{\delta}\right) \nonumber\\
&  =\sum_{\substack{\left(  \gamma,\delta\right)  \in\operatorname*{Comp}%
\nolimits_{n}\times\operatorname*{Comp}\nolimits_{m};\\D\left(  \alpha\right)
\subseteq D\left(  \gamma\right)  \text{ and }D\left(  \beta\right)  \subseteq
D\left(  \delta\right)  }}\underbrace{\dfrac{1}{2^{\ell\left(  \gamma\right)
}}\cdot\dfrac{1}{2^{\ell\left(  \delta\right)  }}}_{=\dfrac{1}{2^{\ell\left(
\gamma\right)  +\ell\left(  \delta\right)  }}=\dfrac{1}{2^{\ell\left(
\gamma\delta\right)  }}}\underbrace{\left(  -1\right)  ^{\ell\left(
\gamma\right)  -\ell\left(  \alpha\right)  }\left(  -1\right)  ^{\ell\left(
\delta\right)  -\ell\left(  \beta\right)  }}_{\substack{=\left(  -1\right)
^{\left(  \ell\left(  \gamma\right)  -\ell\left(  \alpha\right)  \right)
+\left(  \ell\left(  \delta\right)  -\ell\left(  \beta\right)  \right)
}\\=\left(  -1\right)  ^{\left(  \ell\left(  \gamma\right)  +\ell\left(
\delta\right)  \right)  -\left(  \ell\left(  \alpha\right)  +\ell\left(
\beta\right)  \right)  }\\=\left(  -1\right)  ^{\ell\left(  \gamma
\delta\right)  -\ell\left(  \alpha\beta\right)  }}}\underbrace{H_{\gamma
}H_{\delta}}_{=H_{\gamma\delta}}\nonumber\\
&  =\sum_{\substack{\left(  \gamma,\delta\right)  \in\operatorname*{Comp}%
\nolimits_{n}\times\operatorname*{Comp}\nolimits_{m};\\D\left(  \alpha\right)
\subseteq D\left(  \gamma\right)  \text{ and }D\left(  \beta\right)  \subseteq
D\left(  \delta\right)  }}\dfrac{1}{2^{\ell\left(  \gamma\delta\right)  }%
}\left(  -1\right)  ^{\ell\left(  \gamma\delta\right)  -\ell\left(
\alpha\beta\right)  }H_{\gamma\delta}. \label{pf.prop.etastar.concat2.prod3}%
\end{align}
But it is easy to see that every two compositions $\gamma\in
\operatorname*{Comp}\nolimits_{n}$ and $\delta\in\operatorname*{Comp}%
\nolimits_{m}$ satisfy $D\left(  \gamma\delta\right)  =D\left(  \gamma\right)
\cup\left\{  n\right\}  \cup\left(  D\left(  \delta\right)  +n\right)  $,
where $D\left(  \delta\right)  +n$ denotes the set $\left\{  d+n\ \mid\ d\in
D\left(  \delta\right)  \right\}  $. Using this fact, it is easy to see that
the map%
\begin{align*}
&  \left\{  \left(  \gamma,\delta\right)  \in\operatorname*{Comp}%
\nolimits_{n}\times\operatorname*{Comp}\nolimits_{m}\ \mid\ D\left(
\alpha\right)  \subseteq D\left(  \gamma\right)  \text{ and }D\left(
\beta\right)  \subseteq D\left(  \delta\right)  \right\} \\
&  \rightarrow\left\{  \zeta\in\operatorname*{Comp}\nolimits_{n+m}%
\ \mid\ D\left(  \alpha\beta\right)  \subseteq D\left(  \zeta\right)
\right\}  ,\\
\left(  \gamma,\delta\right)   &  \mapsto\gamma\delta
\end{align*}
is well-defined and is a bijection. Hence, we can substitute $\zeta$ for
$\gamma\delta$ in the sum on the right hand side of
(\ref{pf.prop.etastar.concat2.prod3}). Thus,
(\ref{pf.prop.etastar.concat2.prod3}) rewrites as%
\[
\eta_{\alpha}^{\ast}\eta_{\beta}^{\ast}=\sum_{\substack{\zeta\in
\operatorname*{Comp}\nolimits_{n+m};\\D\left(  \alpha\beta\right)  \subseteq
D\left(  \zeta\right)  }}\dfrac{1}{2^{\ell\left(  \zeta\right)  }}\left(
-1\right)  ^{\ell\left(  \zeta\right)  -\ell\left(  \alpha\beta\right)
}H_{\zeta}.
\]
Comparing this with%
\[
\eta_{\alpha\beta}^{\ast}=\sum_{\substack{\zeta\in\operatorname*{Comp}%
\nolimits_{n+m};\\D\left(  \alpha\beta\right)  \subseteq D\left(
\zeta\right)  }}\dfrac{1}{2^{\ell\left(  \zeta\right)  }}\left(  -1\right)
^{\ell\left(  \zeta\right)  -\ell\left(  \alpha\beta\right)  }H_{\zeta
}\ \ \ \ \ \ \ \ \ \ \left(  \text{by the definition of }\eta_{\alpha\beta
}^{\ast}\right)  ,
\]
we obtain $\eta_{\alpha}^{\ast}\eta_{\beta}^{\ast}=\eta_{\alpha\beta}^{\ast}$.
This proves Proposition \ref{prop.etastar.concat2}.
\end{proof}

\begin{corollary}
\label{cor.etastar.concat}Let $\beta_{1},\beta_{2},\ldots,\beta_{k}$ be
finitely many compositions. Then,%
\[
\eta_{\beta_{1}}^{\ast}\eta_{\beta_{2}}^{\ast}\cdots\eta_{\beta_{k}}^{\ast
}=\eta_{\beta_{1}\beta_{2}\cdots\beta_{k}}^{\ast}.
\]

\end{corollary}

\begin{proof}
TODO. (This follows by induction on $k$ using Proposition
\ref{prop.etastar.concat2}.)
\end{proof}

\begin{proof}
[Proof of Proposition \ref{prop.etastar.mult}.]This follows from applying
Corollary \ref{cor.etastar.concat} to the $1$-element compositions $\beta
_{i}=\left(  \alpha_{i}\right)  $ (since $\eta_{n}^{\ast}=\eta_{\left(
n\right)  }^{\ast}$ for each $n>0$).
\end{proof}

Define the formal power series%
\[
H\left(  t\right)  =\sum_{n\geq0}H_{n}t^{n}\in\operatorname*{NSym}\left[
\left[  t\right]  \right]
\]
and
\[
G\left(  t\right)  =\sum_{n\geq1}\eta_{n}^{\ast}t^{n}\in\operatorname*{NSym}%
\left[  \left[  t\right]  \right]  .
\]
Then, it is easy to see that:

\begin{proposition}
\label{prop.Gt-through-Ht}We have%
\[
G\left(  t\right)  =1-\dfrac{1}{1+\dfrac{H\left(  t\right)  -1}{2}}%
=\dfrac{H\left(  t\right)  -1}{H\left(  t\right)  +1}.
\]

\end{proposition}

\begin{proof}
TODO. (Indeed, the first equality sign follows from the geometric series
expansion, and the second is simple manipulation.)
\end{proof}

Consider the comultiplication $\Delta:\operatorname*{NSym}\rightarrow
\operatorname*{NSym}\otimes\operatorname*{NSym}$ of the Hopf algebra
$\operatorname*{NSym}$. The following formula for $\Delta\left(  \eta
_{n}^{\ast}\right)  $ is a dual of Theorem \ref{thm.etaaetab}:

\begin{theorem}
\label{thm.Deltaetastara}Let $n\in\mathbb{N}$. Then,%
\[
\Delta\left(  \eta_{n}^{\ast}\right)  =\sum_{\substack{\beta,\gamma
\in\operatorname*{Comp};\\\left\vert \beta\right\vert +\left\vert
\gamma\right\vert =n;\\\ell\left(  \beta\right)  -\ell\left(  \gamma\right)
\in\left\{  1,-1\right\}  }}\left(  -1\right)  ^{\min\left\{  \ell\left(
\beta\right)  ,\ell\left(  \gamma\right)  \right\}  }\eta_{\beta}^{\ast
}\otimes\eta_{\gamma}^{\ast}.
\]

\end{theorem}

\begin{proof}
From $G\left(  t\right)  =\sum_{n\geq1}\eta_{n}^{\ast}t^{n}$, we obtain%
\begin{align}
G\left(  t\right)  ^{k}  &  =\left(  \sum_{n\geq1}\eta_{n}^{\ast}t^{n}\right)
^{k}=\sum_{n_{1},n_{2},\ldots,n_{k}\geq1}\eta_{n_{1}}^{\ast}\eta_{n_{2}}%
^{\ast}\cdots\eta_{n_{k}}^{\ast}t^{n_{1}+n_{2}+\cdots+n_{k}}\nonumber\\
&  =\sum_{\substack{\beta=\left(  \beta_{1},\beta_{2},\ldots,\beta_{k}\right)
\in\operatorname*{Comp};\\\ell\left(  \beta\right)  =k}}\underbrace{\eta
_{\beta_{1}}^{\ast}\eta_{\beta_{2}}^{\ast}\cdots\eta_{\beta_{k}}^{\ast}%
}_{\substack{=\eta_{\beta}^{\ast}\\\text{(by Proposition
\ref{prop.etastar.mult})}}}\underbrace{t^{\beta_{1}+\beta_{2}+\cdots+\beta
_{k}}}_{=t^{\left\vert \beta\right\vert }}\nonumber\\
&  =\sum_{\substack{\beta=\left(  \beta_{1},\beta_{2},\ldots,\beta_{k}\right)
\in\operatorname*{Comp};\\\ell\left(  \beta\right)  =k}}\eta_{\beta}^{\ast
}t^{\left\vert \beta\right\vert }=\sum_{\substack{\beta\in\operatorname*{Comp}%
;\\\ell\left(  \beta\right)  =k}}\eta_{\beta}^{\ast}t^{\left\vert
\beta\right\vert } \label{pf.thm.Deltaetastara.Gtk}%
\end{align}
for every $k\in\mathbb{N}$.

The comultiplication $\Delta:\operatorname*{NSym}\rightarrow
\operatorname*{NSym}\otimes\operatorname*{NSym}$ induces a $\mathbf{k}%
$-algebra homomorphism%
\[
\Delta_{t}:\operatorname*{NSym}\left[  \left[  t\right]  \right]
\rightarrow\left(  \operatorname*{NSym}\otimes\operatorname*{NSym}\right)
\left[  \left[  t\right]  \right]
\]
that sends each formal power series $\sum_{i\in\mathbb{N}}a_{i}t^{i}$ to
$\sum_{i\in\mathbb{N}}\Delta\left(  a_{i}\right)  t^{i}$. Note that there is a
canonical $\mathbf{k}$-algebra homomorphism%
\begin{align*}
\iota:\operatorname*{NSym}\left[  \left[  t\right]  \right]  \otimes
_{\mathbf{k}\left[  \left[  t\right]  \right]  }\operatorname*{NSym}\left[
\left[  t\right]  \right]   &  \rightarrow\left(  \operatorname*{NSym}%
\otimes\operatorname*{NSym}\right)  \left[  \left[  t\right]  \right]  ,\\
\left(  \sum_{i\in\mathbb{N}}a_{i}t^{i}\right)  \otimes\left(  \sum
_{j\in\mathbb{N}}b_{j}t^{j}\right)   &  \mapsto\left(  \sum_{i\in\mathbb{N}%
}\left(  a_{i}\otimes1\right)  t^{i}\right)  \left(  \sum_{j\in\mathbb{N}%
}\left(  1\otimes b_{j}\right)  t^{j}\right)  .
\end{align*}
Authors often treat $\iota$ as an embedding. We won't, but we will still use
the fact that $\iota$ is a $\mathbf{k}$-algebra homomorphism a lot without saying.

It is easy to see that%
\begin{equation}
\Delta_{t}\left(  H\left(  t\right)  \right)  =\iota\left(  H\left(  t\right)
\otimes H\left(  t\right)  \right)  \label{pf.thm.Deltaetastara.DeltaH}%
\end{equation}
(this follows from \cite[(5.4.2)]{GriRei}).

Define four elements $h_{1}$, $h_{2}$, $g_{1}$ and $g_{2}$ of $\left(
\operatorname*{NSym}\otimes\operatorname*{NSym}\right)  \left[  \left[
t\right]  \right]  $ by%
\begin{align*}
h_{1}  &  =\iota\left(  H\left(  t\right)  \otimes1\right)
\ \ \ \ \ \ \ \ \ \ \text{and}\ \ \ \ \ \ \ \ \ \ h_{2}=\iota\left(  1\otimes
H\left(  t\right)  \right)  \ \ \ \ \ \ \ \ \ \ \text{and}\\
g_{1}  &  =\iota\left(  G\left(  t\right)  \otimes1\right)
\ \ \ \ \ \ \ \ \ \ \text{and}\ \ \ \ \ \ \ \ \ \ g_{2}=\iota\left(  1\otimes
G\left(  t\right)  \right)  .
\end{align*}
The elements $h_{1}$ and $h_{2}$ commute (since $H\left(  t\right)  \otimes1$
and $1\otimes H\left(  t\right)  $ commute). The elements $\dfrac{1}{h_{1}+1}%
$, $\dfrac{1}{h_{2}+1}$ and $\dfrac{1}{h_{1}h_{2}+1}$ are rational functions
in $h_{1}$ and $h_{2}$ and therefore also commute with them (and with each
other). Thus, $h_{1}$, $h_{2}$, $\dfrac{1}{h_{1}+1}$, $\dfrac{1}{h_{2}+1}$ and
$\dfrac{1}{h_{1}h_{2}+1}$ generate a commutative $\mathbf{k}$-subalgebra
$\mathcal{H}$ of $\left(  \operatorname*{NSym}\otimes\operatorname*{NSym}%
\right)  \left[  \left[  t\right]  \right]  $.

Moreover, from $G\left(  t\right)  =\dfrac{H\left(  t\right)  -1}{H\left(
t\right)  +1}$, we obtain
\begin{equation}
g_{1}=\dfrac{h_{1}-1}{h_{1}+1}\ \ \ \ \ \ \ \ \ \ \text{and}%
\ \ \ \ \ \ \ \ \ \ g_{2}=\dfrac{h_{2}-1}{h_{2}+1}.
\label{pf.thm.Deltaetastara.g1g2}%
\end{equation}
Thus, the elements $g_{1}$ and $g_{2}$ also belong to the commutative
$\mathbf{k}$-subalgebra $\mathcal{H}$ generated by $h_{1}$, $h_{2}$,
$\dfrac{1}{h_{1}+1}$, $\dfrac{1}{h_{2}+1}$ and $\dfrac{1}{h_{1}h_{2}+1}$.

The equality (\ref{pf.thm.Deltaetastara.DeltaH}) becomes%
\begin{align}
\Delta_{t}\left(  H\left(  t\right)  \right)   &  =\iota\left(
\underbrace{H\left(  t\right)  \otimes H\left(  t\right)  }_{=\left(  H\left(
t\right)  \otimes1\right)  \left(  1\otimes H\left(  t\right)  \right)
}\right)  =\iota\left(  \left(  H\left(  t\right)  \otimes1\right)  \left(
1\otimes H\left(  t\right)  \right)  \right) \nonumber\\
&  =\underbrace{\iota\left(  H\left(  t\right)  \otimes1\right)  }_{=h_{1}%
}\cdot\underbrace{\iota\left(  1\otimes H\left(  t\right)  \right)  }_{=h_{2}%
}\nonumber\\
&  =h_{1}h_{2}. \label{pf.thm.Deltaetastara.DeltaH2}%
\end{align}
Hence,%
\begin{equation}
\dfrac{\Delta_{t}\left(  H\left(  t\right)  \right)  -1}{\Delta_{t}\left(
H\left(  t\right)  \right)  +1}=\dfrac{h_{1}h_{2}-1}{h_{1}h_{2}+1}%
=\dfrac{g_{1}+g_{2}}{1+g_{1}g_{2}}. \label{pf.thm.Deltaetastara.abab2}%
\end{equation}
(Indeed, the last equality sign can easily be verified by straightforward
computation in the commutative $\mathbf{k}$-algebra $\mathcal{H}$, using the
equalities (\ref{pf.thm.Deltaetastara.g1g2}).)

From $g_{1}=\iota\left(  G\left(  t\right)  \otimes1\right)  $ and
$g_{2}=\iota\left(  1\otimes G\left(  t\right)  \right)  $, we obtain%
\begin{equation}
g_{1}+g_{2}=\iota\left(  G\left(  t\right)  \otimes1\right)  +\iota\left(
1\otimes G\left(  t\right)  \right)  =\iota\left(  G\left(  t\right)
\otimes1+1\otimes G\left(  t\right)  \right)
\label{pf.thm.Deltaetastara.g1plusg2}%
\end{equation}
(since $\iota$ is a $\mathbf{k}$-algebra homomorphism) and%
\begin{align}
g_{1}g_{2}  &  =\iota\left(  G\left(  t\right)  \otimes1\right)  \cdot
\iota\left(  1\otimes G\left(  t\right)  \right)  =\iota\left(
\underbrace{\left(  1\otimes G\left(  t\right)  \right)  \cdot\left(  G\left(
t\right)  \otimes1\right)  }_{=G\left(  t\right)  \otimes G\left(  t\right)
}\right) \nonumber\\
&  \ \ \ \ \ \ \ \ \ \ \left(  \text{since }\iota\text{ is a }\mathbf{k}%
\text{-algebra homomorphism}\right) \nonumber\\
&  =\iota\left(  G\left(  t\right)  \otimes G\left(  t\right)  \right)  .
\label{pf.thm.Deltaetastara.g1g2o}%
\end{align}


Now, Proposition \ref{prop.Gt-through-Ht} yields%
\begin{align*}
\Delta_{t}\left(  G\left(  t\right)  \right)   &  =\Delta_{t}\left(
\dfrac{H\left(  t\right)  -1}{H\left(  t\right)  +1}\right)  =\dfrac
{\Delta_{t}\left(  H\left(  t\right)  \right)  -1}{\Delta_{t}\left(  H\left(
t\right)  \right)  +1}\\
&  \ \ \ \ \ \ \ \ \ \ \left(  \text{since }\Delta_{t}\text{ is a continuous
}\mathbf{k}\text{-algebra homomorphism}\right) \\
&  =\dfrac{g_{1}+g_{2}}{1+g_{1}g_{2}}\ \ \ \ \ \ \ \ \ \ \left(  \text{by
(\ref{pf.thm.Deltaetastara.abab2})}\right) \\
&  =\sum_{i\in\mathbb{N}}\left(  -1\right)  ^{i}\left(  g_{1}g_{2}\right)
^{i}\left(  g_{1}+g_{2}\right) \\
&  =\sum_{i\in\mathbb{N}}\left(  -1\right)  ^{i}\underbrace{\left(
\iota\left(  G\left(  t\right)  \otimes G\left(  t\right)  \right)  \right)
^{i}\cdot\iota\left(  G\left(  t\right)  \otimes1+1\otimes G\left(  t\right)
\right)  }_{\substack{=\iota\left(  \left(  G\left(  t\right)  ^{i}\otimes
G\left(  t\right)  ^{i}\right)  \left(  G\left(  t\right)  \otimes1+1\otimes
G\left(  t\right)  \right)  \right)  \\\text{(since }\iota\text{ is a
}\mathbf{k}\text{-algebra homomorphism)}}}\\
&  \ \ \ \ \ \ \ \ \ \ \left(  \text{by (\ref{pf.thm.Deltaetastara.g1g2o}) and
(\ref{pf.thm.Deltaetastara.g1plusg2})}\right) \\
&  =\sum_{i\in\mathbb{N}}\left(  -1\right)  ^{i}\iota\left(
\underbrace{\left(  G\left(  t\right)  ^{i}\otimes G\left(  t\right)
^{i}\right)  \left(  G\left(  t\right)  \otimes1+1\otimes G\left(  t\right)
\right)  }_{=G\left(  t\right)  ^{i+1}\otimes G\left(  t\right)  ^{i}+G\left(
t\right)  ^{i}\otimes G\left(  t\right)  ^{i+1}}\right) \\
&  =\sum_{i\in\mathbb{N}}\left(  -1\right)  ^{i}\iota\left(  G\left(
t\right)  ^{i+1}\otimes G\left(  t\right)  ^{i}+G\left(  t\right)  ^{i}\otimes
G\left(  t\right)  ^{i+1}\right) \\
&  =\sum_{\substack{i\in\mathbb{N};\\j\in\mathbb{N};\\i-j\in\left\{
1,-1\right\}  }}\left(  -1\right)  ^{\min\left\{  i,j\right\}  }\iota\left(
\underbrace{G\left(  t\right)  ^{i}}_{\substack{=\sum_{\substack{\beta
\in\operatorname*{Comp};\\\ell\left(  \beta\right)  =i}}\eta_{\beta}^{\ast
}t^{\left\vert \beta\right\vert }\\\text{(by (\ref{pf.thm.Deltaetastara.Gtk}%
))}}}\otimes\underbrace{G\left(  t\right)  ^{j}}_{\substack{=\sum
_{\substack{\gamma\in\operatorname*{Comp};\\\ell\left(  \gamma\right)
=j}}\eta_{\gamma}^{\ast}t^{\left\vert \gamma\right\vert }\\\text{(by
(\ref{pf.thm.Deltaetastara.Gtk}))}}}\right) \\
&  =\sum_{\substack{i\in\mathbb{N};\\j\in\mathbb{N};\\i-j\in\left\{
1,-1\right\}  }}\left(  -1\right)  ^{\min\left\{  i,j\right\}  }%
\underbrace{\iota\left(  \left(  \sum_{\substack{\beta\in\operatorname*{Comp}%
;\\\ell\left(  \beta\right)  =i}}\eta_{\beta}^{\ast}t^{\left\vert
\beta\right\vert }\right)  \otimes\left(  \sum_{\substack{\gamma
\in\operatorname*{Comp};\\\ell\left(  \gamma\right)  =j}}\eta_{\gamma}^{\ast
}t^{\left\vert \gamma\right\vert }\right)  \right)  }_{=\sum_{\substack{\beta
\in\operatorname*{Comp};\\\ell\left(  \beta\right)  =i}}\ \ \sum
_{\substack{\gamma\in\operatorname*{Comp};\\\ell\left(  \gamma\right)
=j}}\eta_{\beta}^{\ast}\otimes\eta_{\gamma}^{\ast}t^{\left\vert \beta
\right\vert +\left\vert \gamma\right\vert }}\\
&  =\sum_{\substack{i\in\mathbb{N};\\j\in\mathbb{N};\\i-j\in\left\{
1,-1\right\}  }}\left(  -1\right)  ^{\min\left\{  i,j\right\}  }%
\sum_{\substack{\beta\in\operatorname*{Comp};\\\ell\left(  \beta\right)
=i}}\ \ \sum_{\substack{\gamma\in\operatorname*{Comp};\\\ell\left(
\gamma\right)  =j}}\eta_{\beta}^{\ast}\otimes\eta_{\gamma}^{\ast}t^{\left\vert
\beta\right\vert +\left\vert \gamma\right\vert }\\
&  =\sum_{\substack{\beta,\gamma\in\operatorname*{Comp};\\\ell\left(
\beta\right)  -\ell\left(  \gamma\right)  \in\left\{  1,-1\right\}  }}\left(
-1\right)  ^{\min\left\{  \ell\left(  \beta\right)  ,\ell\left(
\gamma\right)  \right\}  }\eta_{\beta}^{\ast}\otimes\eta_{\gamma}^{\ast
}t^{\left\vert \beta\right\vert +\left\vert \gamma\right\vert }.
\end{align*}
Comparing coefficients of $t^{n}$ here, we obtain%
\[
\Delta\left(  \eta_{n}^{\ast}\right)  =\sum_{\substack{\beta,\gamma
\in\operatorname*{Comp};\\\left\vert \beta\right\vert +\left\vert
\gamma\right\vert =n;\\\ell\left(  \beta\right)  -\ell\left(  \gamma\right)
\in\left\{  1,-1\right\}  }}\left(  -1\right)  ^{\min\left\{  \ell\left(
\beta\right)  ,\ell\left(  \gamma\right)  \right\}  }\eta_{\beta}^{\ast
}\otimes\eta_{\gamma}^{\ast}%
\]
for every positive integer $n$.
\end{proof}

\subsection{The proof of the product rule}

\begin{proof}
[Proof of Theorem \ref{thm.etaaetab}.]TODO. (This needs some fleshing out.)

Let $\alpha=\left(  \alpha_{1},\alpha_{2},\ldots,\alpha_{k}\right)  $ be any
composition. Then,%
\begin{align}
\Delta\left(  \eta_{\alpha}^{\ast}\right)   &  =\Delta\left(  \eta_{\alpha
_{1}}^{\ast}\eta_{\alpha_{2}}^{\ast}\cdots\eta_{\alpha_{k}}^{\ast}\right)
\ \ \ \ \ \ \ \ \ \ \left(  \text{by Proposition \ref{prop.etastar.mult}%
}\right) \nonumber\\
&  =\Delta\left(  \eta_{\alpha_{1}}^{\ast}\right)  \Delta\left(  \eta
_{\alpha_{2}}^{\ast}\right)  \cdots\Delta\left(  \eta_{\alpha_{k}}^{\ast
}\right)  \ \ \ \ \ \ \ \ \ \ \left(  \text{since }\Delta\text{ is a
}\mathbf{k}\text{-algebra homomorphism}\right) \nonumber\\
&  =\left(  \sum_{\substack{\beta_{1},\gamma_{1}\in\operatorname*{Comp}%
;\\\left\vert \beta_{1}\right\vert +\left\vert \gamma_{1}\right\vert
=\alpha_{1};\\\ell\left(  \beta_{1}\right)  -\ell\left(  \gamma_{1}\right)
\in\left\{  1,-1\right\}  }}\left(  -1\right)  ^{\min\left\{  \ell\left(
\beta_{1}\right)  ,\ell\left(  \gamma_{1}\right)  \right\}  }\eta_{\beta_{1}%
}^{\ast}\otimes\eta_{\gamma_{1}}^{\ast}\right) \nonumber\\
&  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \cdot\left(  \sum_{\substack{\beta
_{2},\gamma_{2}\in\operatorname*{Comp};\\\left\vert \beta_{2}\right\vert
+\left\vert \gamma_{2}\right\vert =\alpha_{2};\\\ell\left(  \beta_{2}\right)
-\ell\left(  \gamma_{2}\right)  \in\left\{  1,-1\right\}  }}\left(  -1\right)
^{\min\left\{  \ell\left(  \beta_{2}\right)  ,\ell\left(  \gamma_{2}\right)
\right\}  }\eta_{\beta_{2}}^{\ast}\otimes\eta_{\gamma_{2}}^{\ast}\right)
\nonumber\\
&  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \cdot\cdots\cdot\left(
\sum_{\substack{\beta_{k},\gamma_{k}\in\operatorname*{Comp};\\\left\vert
\beta_{k}\right\vert +\left\vert \gamma_{k}\right\vert =\alpha_{k}%
;\\\ell\left(  \beta_{k}\right)  -\ell\left(  \gamma_{k}\right)  \in\left\{
1,-1\right\}  }}\left(  -1\right)  ^{\min\left\{  \ell\left(  \beta
_{k}\right)  ,\ell\left(  \gamma_{k}\right)  \right\}  }\eta_{\beta_{k}}%
^{\ast}\otimes\eta_{\gamma_{k}}^{\ast}\right) \nonumber\\
&  \ \ \ \ \ \ \ \ \ \ \left(  \text{by Theorem \ref{thm.Deltaetastara}%
}\right) \nonumber\\
&  =\sum_{\substack{\beta_{1},\beta_{2},\ldots,\beta_{k}\in
\operatorname*{Comp};\\\gamma_{1},\gamma_{2},\ldots,\gamma_{k}\in
\operatorname*{Comp};\\\left\vert \beta_{s}\right\vert +\left\vert \gamma
_{s}\right\vert =\alpha_{s}\text{ for each }s;\\\ell\left(  \beta_{s}\right)
-\ell\left(  \gamma_{s}\right)  \in\left\{  1,-1\right\}  \text{ for each }%
s}}\left(  \left(  -1\right)  ^{\min\left\{  \ell\left(  \beta_{1}\right)
,\ell\left(  \gamma_{1}\right)  \right\}  }\eta_{\beta_{1}}^{\ast}\otimes
\eta_{\gamma_{1}}^{\ast}\right) \nonumber\\
&  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \cdot\left(  \left(  -1\right)
^{\min\left\{  \ell\left(  \beta_{2}\right)  ,\ell\left(  \gamma_{2}\right)
\right\}  }\eta_{\beta_{2}}^{\ast}\otimes\eta_{\gamma_{2}}^{\ast}\right)
\cdot\cdots\cdot\left(  \left(  -1\right)  ^{\min\left\{  \ell\left(
\beta_{k}\right)  ,\ell\left(  \gamma_{k}\right)  \right\}  }\eta_{\beta_{k}%
}^{\ast}\otimes\eta_{\gamma_{k}}^{\ast}\right) \nonumber\\
&  =\sum_{\substack{\beta_{1},\beta_{2},\ldots,\beta_{k}\in
\operatorname*{Comp};\\\gamma_{1},\gamma_{2},\ldots,\gamma_{k}\in
\operatorname*{Comp};\\\left\vert \beta_{s}\right\vert +\left\vert \gamma
_{s}\right\vert =\alpha_{s}\text{ for each }s;\\\ell\left(  \beta_{s}\right)
-\ell\left(  \gamma_{s}\right)  \in\left\{  1,-1\right\}  \text{ for each }%
s}}\left(  -1\right)  ^{\min\left\{  \ell\left(  \beta_{1}\right)
,\ell\left(  \gamma_{1}\right)  \right\}  +\min\left\{  \ell\left(  \beta
_{2}\right)  ,\ell\left(  \gamma_{2}\right)  \right\}  +\cdots+\min\left\{
\ell\left(  \beta_{k}\right)  ,\ell\left(  \gamma_{k}\right)  \right\}
}\nonumber\\
&  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \underbrace{\left(  \eta_{\beta
_{1}}^{\ast}\eta_{\beta_{2}}^{\ast}\cdots\eta_{\beta_{k}}^{\ast}\right)
}_{\substack{=\eta_{\beta_{1}\beta_{2}\cdots\beta_{k}}^{\ast}\\\text{(by
Corollary \ref{cor.etastar.concat})}}}\otimes\underbrace{\left(  \eta
_{\gamma_{1}}^{\ast}\eta_{\gamma_{2}}^{\ast}\cdots\eta_{\gamma_{k}}^{\ast
}\right)  }_{\substack{=\eta_{\gamma_{1}\gamma_{2}\cdots\gamma_{k}}^{\ast
}\\\text{(by Corollary \ref{cor.etastar.concat})}}}\nonumber\\
&  =\sum_{\substack{\beta_{1},\beta_{2},\ldots,\beta_{k}\in
\operatorname*{Comp};\\\gamma_{1},\gamma_{2},\ldots,\gamma_{k}\in
\operatorname*{Comp};\\\left\vert \beta_{s}\right\vert +\left\vert \gamma
_{s}\right\vert =\alpha_{s}\text{ for each }s;\\\ell\left(  \beta_{s}\right)
-\ell\left(  \gamma_{s}\right)  \in\left\{  1,-1\right\}  \text{ for each }%
s}}\left(  -1\right)  ^{\min\left\{  \ell\left(  \beta_{1}\right)
,\ell\left(  \gamma_{1}\right)  \right\}  +\min\left\{  \ell\left(  \beta
_{2}\right)  ,\ell\left(  \gamma_{2}\right)  \right\}  +\cdots+\min\left\{
\ell\left(  \beta_{k}\right)  ,\ell\left(  \gamma_{k}\right)  \right\}
}\nonumber\\
&  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \eta_{\beta_{1}\beta_{2}\cdots
\beta_{k}}^{\ast}\otimes\eta_{\gamma_{1}\gamma_{2}\cdots\gamma_{k}}^{\ast
}\nonumber
\end{align}%
\begin{align}
&  =\sum_{\delta,\varepsilon\in\operatorname*{Comp}}\sum_{\substack{\beta
_{1},\beta_{2},\ldots,\beta_{k}\in\operatorname*{Comp};\\\gamma_{1},\gamma
_{2},\ldots,\gamma_{k}\in\operatorname*{Comp};\\\beta_{1}\beta_{2}\cdots
\beta_{k}=\delta;\\\gamma_{1}\gamma_{2}\cdots\gamma_{k}=\varepsilon
;\\\left\vert \beta_{s}\right\vert +\left\vert \gamma_{s}\right\vert
=\alpha_{s}\text{ for each }s;\\\ell\left(  \beta_{s}\right)  -\ell\left(
\gamma_{s}\right)  \in\left\{  1,-1\right\}  \text{ for each }s}}\left(
-1\right)  ^{\min\left\{  \ell\left(  \beta_{1}\right)  ,\ell\left(
\gamma_{1}\right)  \right\}  +\min\left\{  \ell\left(  \beta_{2}\right)
,\ell\left(  \gamma_{2}\right)  \right\}  +\cdots+\min\left\{  \ell\left(
\beta_{k}\right)  ,\ell\left(  \gamma_{k}\right)  \right\}  }\nonumber\\
&  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \eta_{\delta}^{\ast}\otimes
\eta_{\varepsilon}^{\ast}. \label{pf.thm.etaaetab.5}%
\end{align}
Now, let us take a closer look at the inner sum on the right hand side. For a
given pair $\left(  \delta,\varepsilon\right)  $ of compositions, what are the
pairs%
\[
\left(  \left(  \beta_{1},\beta_{2},\ldots,\beta_{k}\right)  ,\left(
\gamma_{1},\gamma_{2},\ldots,\gamma_{k}\right)  \right)
\]
satisfying the six conditions%
\begin{align*}
\beta_{1},\beta_{2},\ldots,\beta_{k}  &  \in\operatorname*{Comp}%
;\ \ \ \ \ \ \ \ \ \ \gamma_{1},\gamma_{2},\ldots,\gamma_{k}\in
\operatorname*{Comp};\\
\beta_{1}\beta_{2}\cdots\beta_{k}  &  =\delta;\ \ \ \ \ \ \ \ \ \ \gamma
_{1}\gamma_{2}\cdots\gamma_{k}=\varepsilon;\\
\left\vert \beta_{s}\right\vert +\left\vert \gamma_{s}\right\vert  &
=\alpha_{s}\text{ for each }s;\\
\ell\left(  \beta_{s}\right)  -\ell\left(  \gamma_{s}\right)   &  \in\left\{
1,-1\right\}  \text{ for each }s
\end{align*}
? I claim that they are in bijection with the liminal stufflers $f$ for
$\delta$ and $\varepsilon$ that satisfy $\operatorname*{wt}\left(  f\right)
=\alpha$. Indeed, if we write the compositions $\delta$ and $\varepsilon$ as
$\delta=\left(  \delta_{1},\delta_{2},\ldots,\delta_{\ell}\right)  $ and
$\varepsilon=\left(  \varepsilon_{1},\varepsilon_{2},\ldots,\varepsilon
_{m}\right)  $, and if we define $D$ as in Definition \ref{def.stuffler}, then
the bijection sends any such liminal stuffler $f$ to the pair
\[
\left(  \left(  \beta_{1},\beta_{2},\ldots,\beta_{k}\right)  ,\left(
\gamma_{1},\gamma_{2},\ldots,\gamma_{k}\right)  \right)  ,
\]
where
\begin{align*}
\beta_{s}  &  =\left(  \text{the composition consisting of the }\delta
_{u}\text{ for all }u\in\left[  \ell\right]  \text{ satisfying }f\left(
p_{u}\right)  =s\right. \\
&  \ \ \ \ \ \ \ \ \ \ \left.  \text{(in the order of increasing }%
u\text{)}\right) \\
\gamma_{s}  &  =\left(  \text{the composition consisting of the }%
\varepsilon_{v}\text{ for all }v\in\left[  m\right]  \text{ satisfying
}f\left(  q_{v}\right)  =s\right. \\
&  \ \ \ \ \ \ \ \ \ \ \left.  \text{(in the order of increasing }%
v\text{)}\right)
\end{align*}
(We are here using the fact that our liminal stuffler $f$ must necessarily be
a map from $D$ to $\left\{  1<2<\cdots<k\right\}  $, because its weight
$\operatorname*{wt}\left(  f\right)  =\alpha$ is a composition of length $k$.)

Using this bijection, we can rewrite (\ref{pf.thm.etaaetab.5}) as%
\begin{equation}
\Delta\left(  \eta_{\alpha}^{\ast}\right)  =\sum_{\delta,\varepsilon
\in\operatorname*{Comp}}\ \ \sum_{\substack{f\text{ is a liminal
stuffler}\\\text{for }\delta\text{ and }\varepsilon;\\\operatorname*{wt}%
\left(  f\right)  =\alpha}}\left(  -1\right)  ^{\operatorname*{loss}\left(
f\right)  }\eta_{\delta}^{\ast}\otimes\eta_{\varepsilon}^{\ast}.
\label{pf.thm.etaaetab.at}%
\end{equation}
(Here, we have used the fact that if our bijection sends a liminal stuffler
$f$ to a pair $\left(  \left(  \beta_{1},\beta_{2},\ldots,\beta_{k}\right)
,\left(  \gamma_{1},\gamma_{2},\ldots,\gamma_{k}\right)  \right)  $, then%
\[
\min\left\{  \ell\left(  \beta_{1}\right)  ,\ell\left(  \gamma_{1}\right)
\right\}  +\min\left\{  \ell\left(  \beta_{2}\right)  ,\ell\left(  \gamma
_{2}\right)  \right\}  +\cdots+\min\left\{  \ell\left(  \beta_{k}\right)
,\ell\left(  \gamma_{k}\right)  \right\}  =\operatorname*{loss}\left(
f\right)  .
\]
This is easily seen from the definition of the bijection and of the loss.)

Dualizing the equality (\ref{pf.thm.etaaetab.at}), we find precisely the claim
of Theorem \ref{thm.etaaetab}. (This needs some formalization.)
\end{proof}

\subsection{The coproduct of $\eta_{\alpha}$}

Consider the coproduct $\Delta:\operatorname*{QSym}\rightarrow
\operatorname*{QSym}\otimes\operatorname*{QSym}$ of the Hopf algebra
$\operatorname*{QSym}$. (See \cite[\S 5.1]{GriRei} for its definition.) We
claim the following simple formula for $\Delta\left(  \eta_{\alpha}\right)  $:

\begin{theorem}
\label{thm.Delta-eta}Let $\alpha\in\operatorname*{Comp}$. Then,%
\[
\Delta\left(  \eta_{\alpha}\right)  =\sum_{\substack{\beta,\gamma
\in\operatorname*{Comp};\\\alpha=\beta\gamma}}\eta_{\beta}\otimes\eta_{\gamma
}.
\]

\end{theorem}

This generalizes \cite[Corollary 2.7]{Hsiao07}.

\begin{proof}
[Proof of Theorem \ref{thm.Delta-eta}.]TODO. (This follows by dualizing
Proposition \ref{prop.etastar.concat2}.)
\end{proof}

\section{The liminal stuffle algebra}

Here is a different way of rewriting Theorem \ref{thm.etaaetab}.

Let $\mathcal{F}$ be the free $\mathbf{k}$-algebra with generators
$x_{1},x_{2},x_{3},\ldots$. It has a basis consisting of all words over the
alphabet $\left\{  x_{1},x_{2},x_{3},\ldots\right\}  $; these words are in
bijection with the compositions.

For any $k\geq0$, we let $r_{k}:\mathcal{F}\rightarrow\mathcal{F}$ be the
linear operator defined recursively by%
\begin{align*}
r_{k}\left(  1\right)   &  =0;\\
r_{k}\left(  x_{i}w\right)   &  =x_{i+k}w\ \ \ \ \ \ \ \ \ \ \text{for each
}i\geq1\text{ and any word }w.
\end{align*}
(Thus, explicitly: $r_{k}$ sends $1$ to $0$, and transforms any nonempty word
by adding $k$ to the subscript of its first letter. For example, $r_{k}\left(
x_{a}x_{b}x_{c}\right)  =x_{a+k}x_{b}x_{c}$.)

Let $\#:\mathcal{F}\times\mathcal{F}\rightarrow\mathcal{F}$ be the bilinear
map on $\mathcal{F}$ defined recursively by%
\begin{align*}
1\#w  &  =w\ \ \ \ \ \ \ \ \ \ \text{for any word }w;\\
w\#1  &  =w\ \ \ \ \ \ \ \ \ \ \text{for any word }w;\\
\left(  x_{i}u\right)  \#\left(  x_{j}v\right)   &  =x_{i}\left(  u\#\left(
x_{j}v\right)  \right)  +x_{j}\left(  \left(  x_{i}u\right)  \#v\right)
-r_{i+j}\left(  u\#v\right)  .
\end{align*}
I call this bilinear map $\#$ the \emph{liminal stuffle}. Thus, if
$u=x_{\alpha_{1}}x_{\alpha_{2}}\cdots x_{\alpha_{\ell}}$ and $v=x_{\beta_{1}%
}x_{\beta_{2}}\cdots x_{\beta_{m}}$ are two words in $\mathcal{F}$, then%
\[
u\#v=\sum_{\substack{f\text{ is a liminal stuffler}\\\text{for }\alpha\text{
and }\beta}}\left(  -1\right)  ^{\operatorname*{loss}\left(  f\right)
}x_{\operatorname*{wt}\left(  f\right)  },
\]
where we set $x_{\gamma}=x_{\gamma_{1}}x_{\gamma_{2}}\cdots x_{\gamma_{k}}$
for every composition $\gamma=\left(  \gamma_{1},\gamma_{2},\ldots,\gamma
_{k}\right)  $. (This needs to be formally proven.)

Let $\operatorname*{eta}:\mathcal{F}\rightarrow\operatorname*{QSym}$ be the
$\mathbf{k}$-linear map that sends $x_{\alpha_{1}}x_{\alpha_{2}}\cdots
x_{\alpha_{k}}$ to $\eta_{\alpha}$ for each composition $\alpha=\left(
\alpha_{1},\alpha_{2},\ldots,\alpha_{k}\right)  $. Then, the claim of Theorem
\ref{thm.etaaetab} is that $\operatorname*{eta}\left(  u\#v\right)
=\operatorname*{eta}u\cdot\operatorname*{eta}v$.

\begin{proposition}
The liminal stuffle $\#$ on $\mathcal{F}$ is commutative and associative and
has identity $1$.
\end{proposition}

\begin{proof}
Only associativity is nontrivial. It follows from Theorem \ref{thm.etaaetab},
since $\operatorname*{eta}$ is injective when $2$ is invertible in
$\mathbf{k}$. Is there a direct proof? I haven't found one so far, but it is
likely an induction argument.
\end{proof}

Thus, $\left(  \mathcal{F},\#\right)  $ is a $\mathbf{k}$-algebra with unity
$1$. Hence, $\operatorname*{eta}$ is a $\mathbf{k}$-algebra homomorphism from
$\left(  \mathcal{F},\#\right)  $ to $\operatorname*{QSym}$.

Let $\Delta:\mathcal{F}\rightarrow\mathcal{F}\otimes\mathcal{F}$ be the
$\mathbf{k}$-linear map that sends each word $w_{1}w_{2}\cdots w_{n}$ to
$\sum_{i=0}^{n}w_{1}w_{2}\cdots w_{i}\otimes w_{i+1}w_{i+2}\cdots w_{n}$. This
map $\Delta$ is called the \textit{deconcatenation coproduct} (or the
\textit{cut coproduct}). This coproduct turns $\mathcal{F}$ into a Hopf
algebra (with counit $\varepsilon:\mathcal{F}\rightarrow\mathbf{k}$ sending
each word $w_{1}w_{2}\cdots w_{n}$ to $%
\begin{cases}
1, & \text{if }n=0;\\
0, & \text{if }n>0
\end{cases}
$). The map $\operatorname*{eta}:\mathcal{F}\rightarrow\operatorname*{QSym}$
is then easily seen to be a $\mathbf{k}$-coalgebra homomorphism (by Theorem
\ref{thm.Delta-eta}).

The liminal stuffle $\#$ on $\mathcal{F}$ respects the deconcatenation
coproduct $\Delta$ of $\mathcal{F}$, in the following sense:

\begin{proposition}
We have $\Delta\left(  u\right)  \#\Delta\left(  v\right)  =\Delta\left(
u\#v\right)  $ for any $u,v\in\mathcal{F}$. Here, the \textquotedblleft%
$\#$\textquotedblright\ sign on the left hand side stands for the
multiplication in the $\mathbf{k}$-algebra $\left(  \mathcal{F},\#\right)
\otimes\left(  \mathcal{F},\#\right)  $ (so, explicitly, is given by $\left(
u_{1}\otimes u_{2}\right)  \#\left(  v_{1}\otimes v_{2}\right)  =\left(
u_{1}\#v_{1}\right)  \otimes\left(  u_{2}\#v_{2}\right)  $ for any
$u_{1},u_{2},v_{1},v_{2}\in\mathcal{F}$).
\end{proposition}

\begin{proof}
This follows from the corresponding property of $\operatorname*{QSym}$, since
$\operatorname*{eta}:\mathcal{F}\rightarrow\operatorname*{QSym}$ is an
injective $\mathbf{k}$-coalgebra homomorphism when $2$ is invertible in
$\mathbf{k}$.

Probably an inductive proof also exists.
\end{proof}

\section{The original problem}

\subsection{$2$-phobic compositions}

\begin{definition}
We say that a composition $\alpha$ is $2$\textit{-phobic} if none of the
entries of $\alpha$ equals $2$.
\end{definition}

\begin{proposition}
\label{prop.2-phobic.span-eta-alg}The span $\operatorname*{span}\left(
\eta_{\alpha}\right)  _{\alpha\text{ is a }2\text{-phobic composition}}$ is a
$\mathbf{k}$-subalgebra of $\operatorname*{QSym}$.
\end{proposition}

\begin{proof}
This follows from Theorem \ref{thm.etaaetab}, because if $\alpha$ and $\beta$
are $2$-phobic compositions, then $\operatorname*{wt}\left(  f\right)  $ is a
$2$-phobic composition whenever $f$ is a limited stuffler for $\alpha$ and
$\beta$. (Note that this is a special property of $2$. Nothing like this holds
for $4$-phobic compositions.)
\end{proof}

\begin{remark}
TODO! Proposition \ref{prop.2-phobic.span-eta-alg} seems to be the same as
\cite[Theorem 5.7]{BMSW00}, if my $\eta_{\alpha}$ are (proportional to) their
$\theta_{\alpha}$ (possibly for different $\alpha$). Check this and say this.
\end{remark}

\subsection{The watered-down original problem}

Now, we define another family of quasisymmetric functions (not a basis this time):

\begin{definition}
\label{def.Knbar}Let $n\in\mathbb{N}$, and let $\Lambda\subseteq\left[
n\right]  $. Then, we set%
\[
\overline{K}_{n,\Lambda}=\sum_{\substack{\left(  g_{1},g_{2},\ldots
,g_{n}\right)  \in\left\{  1,2,3,\ldots\right\}  ^{n};\\g_{1}\leq g_{2}%
\leq\cdots\leq g_{n};\\\text{no }i\in\Lambda\text{ satisfies }g_{i-1}%
=g_{i}=g_{i+1}\\\text{(where we set }g_{0}=0\text{ and }g_{n+1}=\infty
\text{)}}}2^{\left\vert \left\{  g_{1},g_{2},\ldots,g_{n}\right\}  \right\vert
}x_{g_{1}}x_{g_{2}}\cdots x_{g_{n}}.
\]

\end{definition}

\begin{theorem}
\label{thm.Knbar.alg}\textbf{(a)} These $\overline{K}_{n,\Lambda}$ are
quasisymmetric functions.

\textbf{(b)} The span $\operatorname*{span}\left(  \overline{K}_{n,\Lambda
}\right)  _{n\in\mathbb{N};\ \Lambda\subseteq\left[  n\right]  }$ is a
$\mathbf{k}$-subalgebra of $\operatorname*{QSym}$.
\end{theorem}

\begin{proof}
[Proof of Theorem \ref{thm.Knbar.alg}.]Set%
\[
\overline{L}_{n,\Lambda}=\sum_{\substack{\left(  g_{1},g_{2},\ldots
,g_{n}\right)  \in\left\{  1,2,3,\ldots\right\}  ^{n};\\g_{1}\leq g_{2}%
\leq\cdots\leq g_{n};\\\text{\textbf{each} }i\in\Lambda\text{ satisfies
}g_{i-1}=g_{i}=g_{i+1}\\\text{(where we set }g_{0}=0\text{ and }g_{n+1}%
=\infty\text{)}}}2^{\left\vert \left\{  g_{1},g_{2},\ldots,g_{n}\right\}
\right\vert }x_{g_{1}}x_{g_{2}}\cdots x_{g_{n}}%
\]
for each $\Lambda\subseteq\left[  n\right]  $. Then, a standard
inclusion/exclusion computation yields%
\[
\overline{K}_{n,\Lambda}=\sum_{M\subseteq\Lambda}\left(  -1\right)
^{\left\vert M\right\vert }\overline{L}_{n,M}%
\]
for each $\Lambda\subseteq\left[  n\right]  $. Thus,%
\[
\operatorname*{span}\left(  \overline{K}_{n,\Lambda}\right)  _{n\in
\mathbb{N};\ \Lambda\subseteq\left[  n\right]  }=\operatorname*{span}\left(
\overline{L}_{n,\Lambda}\right)  _{n\in\mathbb{N};\ \Lambda\subseteq\left[
n\right]  }%
\]
(by the standard triangularity argument).

But $\overline{L}_{n,\Lambda}$ is $0$ if $1$ or $n$ belongs to $\Lambda$
(since neither $g_{0}=g_{1}$ nor $g_{n}=g_{n+1}$ can hold), and otherwise
equals $\eta_{\alpha}$ for the $2$-phobic composition $\alpha$ defined by
$\left[  n-1\right]  \setminus D\left(  \alpha\right)  =\Lambda\cup\left(
\Lambda-1\right)  $. Conversely, for each $2$-phobic composition $\alpha$, we
can find a $\Lambda\subseteq\left[  n\right]  $ containing neither $1$ nor $n$
that satisfies $\overline{L}_{n,\Lambda}=\eta_{\alpha}$. Thus, the families
$\left(  \overline{L}_{n,\Lambda}\right)  _{n\in\mathbb{N};\ \Lambda
\subseteq\left[  n\right]  }$ and $\left(  \eta_{\alpha}\right)
_{\alpha\text{ is a }2\text{-phobic composition}}$ contain the same nonzero
elements. Hence,%
\[
\operatorname*{span}\left(  \overline{L}_{n,\Lambda}\right)  _{n\in
\mathbb{N};\ \Lambda\subseteq\left[  n\right]  }=\operatorname*{span}\left(
\eta_{\alpha}\right)  _{\alpha\text{ is a }2\text{-phobic composition}}.
\]
Hence,%
\begin{align*}
\operatorname*{span}\left(  \overline{K}_{n,\Lambda}\right)  _{n\in
\mathbb{N};\ \Lambda\subseteq\left[  n\right]  }  &  =\operatorname*{span}%
\left(  \overline{L}_{n,\Lambda}\right)  _{n\in\mathbb{N};\ \Lambda
\subseteq\left[  n\right]  }\\
&  =\operatorname*{span}\left(  \eta_{\alpha}\right)  _{\alpha\text{ is a
}2\text{-phobic composition}}.
\end{align*}
Thus, Theorem \ref{thm.Knbar.alg} \textbf{(a)} follows immediately (since
$\operatorname*{span}\left(  \eta_{\alpha}\right)  _{\alpha\text{ is a
}2\text{-phobic composition}}\subseteq\operatorname*{QSym}$), and Theorem
\ref{thm.Knbar.alg} \textbf{(b)} follows from Proposition
\ref{prop.2-phobic.span-eta-alg}.
\end{proof}

\subsection{Note on the $\Phi$ and $\Psi$ bases}

\begin{remark}
Consider the $\Phi$ and $\Psi$ bases of $\operatorname*{NSym}%
\nolimits_{\mathbb{Q}}$ from \cite{GKLLRT94}, and consider their dual bases
$\phi$ and $\psi$ of $\operatorname*{QSym}\nolimits_{\mathbb{Q}}$
(\cite{BDHMN17}). Then, over $\mathbb{Q}$, we have%
\begin{align*}
\operatorname*{span}\left(  \phi_{\alpha}\right)  _{\alpha\text{ is a
}2\text{-phobic composition}}  &  =\operatorname*{span}\left(  \psi_{\alpha
}\right)  _{\alpha\text{ is a }2\text{-phobic composition}}\\
&  =\operatorname*{span}\left(  \eta_{\alpha}\right)  _{\alpha\text{ is a
}2\text{-phobic composition}}.
\end{align*}
This follows from the fact that $\Phi_{2}=\Psi_{2}=4\eta_{2}^{\ast}$.
\end{remark}

\subsection{The original problem}

Here comes the original problem (\cite[Question 2.51]{Grinbe18}) that all the
above was invented to solve.

Let $\mathcal{N}$ be the totally ordered set $\left\{  0,1,2,\ldots\right\}
\cup\left\{  \infty\right\}  $, with total order given by $0\prec1\prec
2\prec\cdots\prec\infty$.

Let $\operatorname*{Pow}\mathcal{N}$ be the ring of formal power series
$\mathbb{Z}\left[  \left[  x_{0},x_{1},x_{2},\ldots,x_{\infty}\right]
\right]  $.

Let $n\in\mathbb{N}$. For any map $g:\left[  n\right]  \rightarrow\mathcal{N}%
$, we define a subset $\operatorname*{FE}\left(  g\right)  $ of $\left[
n\right]  $ as follows:%
\begin{align*}
\operatorname*{FE}\left(  g\right)   &  =\left\{  \min\left(  g^{-1}\left(
h\right)  \right)  \ \mid\ h\in\left\{  1,2,3,\ldots,\infty\right\}  \text{
with }g^{-1}\left(  h\right)  \neq\varnothing\right\} \\
&  \ \ \ \ \ \ \ \ \ \ \cup\left\{  \max\left(  g^{-1}\left(  h\right)
\right)  \ \mid\ h\in\left\{  0,1,2,3,\ldots\right\}  \text{ with }%
g^{-1}\left(  h\right)  \neq\varnothing\right\}  .
\end{align*}
In other words, $\operatorname*{FE}\left(  g\right)  $ is the set comprising
the smallest elements of all nonempty fibers of $g$ except for $g^{-1}\left(
0\right)  $ as well as the largest elements of all nonempty fibers of $g$
except for $g^{-1}\left(  \infty\right)  $. We shall refer to the elements of
$\operatorname*{FE}\left(  g\right)  $ as the \emph{fiber-ends} of $g$.

If $\Lambda$ is any subset of $\left[  n\right]  $, then we define a power
series $K_{n,\Lambda}^{\mathcal{Z}}\in\operatorname*{Pow}\mathcal{N}$ by%
\begin{align*}
K_{n,\Lambda}^{\mathcal{Z}}  &  =\sum_{\substack{g:\left[  n\right]
\rightarrow\mathcal{N}\text{ is}\\\text{weakly increasing;}\\\Lambda
\subseteq\operatorname*{FE}\left(  g\right)  }}2^{\left\vert g\left(  \left[
n\right]  \right)  \cap\left\{  1,2,3,\ldots\right\}  \right\vert }%
\mathbf{x}_{g}\\
&  =\sum_{\substack{g:\left[  n\right]  \rightarrow\mathcal{N}\text{ is weakly
increasing;}\\\text{no }i\in\Lambda\text{ satisfies }g\left(  i-1\right)
=g\left(  i\right)  =g\left(  i+1\right)  \\\text{(where we set }g\left(
0\right)  =0\text{ and }g\left(  n+1\right)  =\infty\text{)}}}2^{\left\vert
g\left(  \left[  n\right]  \right)  \cap\left\{  1,2,3,\ldots\right\}
\right\vert }\mathbf{x}_{g}\\
&  =\sum_{\substack{\left(  g_{1},g_{2},\ldots,g_{n}\right)  \in
\mathcal{N}^{n};\\0\preccurlyeq g_{1}\preccurlyeq g_{2}\preccurlyeq
\cdots\preccurlyeq g_{n}\preccurlyeq\infty;\\\text{no }i\in\Lambda\text{
satisfies }g_{i-1}=g_{i}=g_{i+1}\\\text{(where we set }g_{0}=0\text{ and
}g_{n+1}=\infty\text{)}}}2^{\left\vert \left\{  g_{1},g_{2},\ldots
,g_{n}\right\}  \cap\left\{  1,2,3,\ldots\right\}  \right\vert }x_{g_{1}%
}x_{g_{2}}\cdots x_{g_{n}},
\end{align*}
where $\mathbf{x}_{g}:=x_{g\left(  1\right)  }x_{g\left(  2\right)  }\cdots
x_{g\left(  n\right)  }$.

\begin{question}
\label{quest.ori}Is the span $\operatorname*{span}\left(  K_{n,\Lambda
}^{\mathcal{Z}}\right)  _{n\in\mathbb{N};\ \Lambda\subseteq\left[  n\right]
}$ a $\mathbb{Z}$-subalgebra of $\operatorname*{Pow}\mathcal{N}$ ?
\end{question}

In other words, is it true that $K_{n,\Lambda}^{\mathcal{Z}}\cdot K_{m,\Omega
}^{\mathcal{Z}}$ is a $\mathbb{Z}$-linear combination of $K_{n+m,\Xi
}^{\mathcal{Z}}$'s whenever $\Lambda\subseteq\left[  n\right]  $ and
$\Omega\subseteq\left[  m\right]  $ are sets?

In \cite[Corollary 2.42]{Grinbe18}, I have shown a formula for rewriting a
product of the form $K_{n,\Lambda}^{\mathcal{Z}}\cdot K_{m,\Omega
}^{\mathcal{Z}}$ as a $\mathbb{Z}$-linear combination of $K_{n+m,\Xi
}^{\mathcal{Z}}$'s when $\Lambda\subseteq\left[  n\right]  $ and
$\Omega\subseteq\left[  m\right]  $ are \textbf{lacunar nonempty} sets. But
can it also be rewritten in such a way if $\Lambda$ and $\Omega$ are arbitrary
subsets of $\left[  n\right]  $ and $\left[  m\right]  $ ? Computations with
SageMath suggest that the answer is \textquotedblleft yes\textquotedblright.
For example,%
\begin{align*}
K_{2,\left\{  1,2\right\}  }^{\mathcal{Z}}\cdot K_{1,\left\{  1\right\}
}^{\mathcal{Z}}  &  =K_{3,\left\{  2\right\}  }^{\mathcal{Z}}+2\cdot
K_{3,\left\{  1,3\right\}  }^{\mathcal{Z}}\ \ \ \ \ \ \ \ \ \ \text{and}\\
K_{2,\varnothing}^{\mathcal{Z}}\cdot K_{1,\left\{  1\right\}  }^{\mathcal{Z}}
&  =K_{3,\varnothing}^{\mathcal{Z}}+K_{3,\left\{  2\right\}  }^{\mathcal{Z}%
}+K_{3,\left\{  1,3\right\}  }^{\mathcal{Z}}=K_{3,\left\{  1\right\}
}^{\mathcal{Z}}+K_{3,\left\{  2\right\}  }^{\mathcal{Z}}+K_{3,\left\{
3\right\}  }^{\mathcal{Z}}.
\end{align*}
But \textbf{in general, the coefficients cannot be taken to be nonnegative}!

Nevertheless, do they have combinatorial interpretations?

Note that the $\mathbb{Q}$-linear span of the $K_{n+m,\Xi}^{\mathcal{Z}}$'s
for all $\Xi\subseteq\left[  n+m\right]  $ is (generally) larger than that of
the $K_{n+m,\Xi}^{\mathcal{Z}}$'s with $\Xi$ lacunar nonempty. It has
dimension \href{https://oeis.org/A005251}{Sequence A005251 in the OEIS} (I believe):

\texttt{a(0) = 0, a(1) = a(2) = a(3) = 1;}

\texttt{thereafter, a(n) = a(n-1) + a(n-2) + a(n-4).}

Theorem \ref{thm.Knbar.alg} \textbf{(b)} shows that Question \ref{quest.ori}
has a positive answer if we set the indeterminates $x_{0}$ and $x_{\infty}$ to
$0$. Indeed, the $\overline{K}_{n,\Lambda}$ from Definition \ref{def.Knbar}
can be rewritten as follows:%
\[
\overline{K}_{n,\Lambda}=K_{n,\Lambda}^{\mathcal{Z}}\mid_{x_{0}=0\text{ and
}x_{\infty}=0}.
\]


Now let us try to go back to the original Question \ref{quest.ori}, where $0$
and $\infty$ are allowed as $g_{i}$-values (and $x_{0}$ and $x_{\infty}$ are
not set to $0$).

Can we transform the duality argument (from the proof of Theorem
\ref{thm.etaaetab}) into a \textquotedblleft Cauchy kernel\textquotedblright%
\ style computation in $\operatorname*{NSym}\nolimits_{\mathbb{Q}}\left[
\left[  x_{0},x_{1},x_{2},\ldots,x_{\infty}\right]  \right]  $ ?

\begin{thebibliography}{99999999}                                                                                         %


\bibitem[AgBeSo14]{AgBeSo14}%
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\bibitem[BDHMN17]{BDHMN17}\href{https://arxiv.org/abs/1710.11613v1}{Cristina
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\bibitem[BMSW00]{BMSW00}\href{https://arxiv.org/abs/math/0002073v2}{Nantel
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\bibitem[GKLLRT94]{GKLLRT94}%
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Lascoux, B. Leclerc, V. S. Retakh, J.-Y. Thibon, \textit{Noncommutative
symmetric functions}, arXiv:hep-th/9407124v1}.

\bibitem[Grinbe14]{Gri-dimm}Darij Grinberg, \textit{Dual immaculate creation
operators and a dendriform algebra structure on the quasisymmetric functions},
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\bibitem[Grinbe15]{Grinbe15}\href{https://arxiv.org/abs/1509.08355v3}{Darij
Grinberg, \textit{Double posets and the antipode of QSym}, arXiv:1509.08355v3}.

\bibitem[Grinbe18]{Grinbe18}\href{https://arxiv.org/abs/1806.04114v3}{Darij
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\bibitem[Grinbe20]{Grinbe20}Darij Grinberg, \textit{Enumerative Combinatorics:
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\bibitem[GriRei20]{GriRei}\href{https://arxiv.org/abs/1409.8356v7}{Darij
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\end{thebibliography}


\end{document}