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\author{Darij Grinberg (UMN)
}
\title[Shuffle-compatibility and exterior peaks]{Ideals of QSym, shuffle-compatibility and exterior peaks}
\date{8 March 2018 \\ University of Illinois at Urbana-Champaign}

\begin{document}

\frame{\titlepage
\textbf{slides: \red \url{http://www.cip.ifi.lmu.de/~grinberg/algebra/urbana18b.pdf}} \\
\textbf{paper: \red \url{http://www.cip.ifi.lmu.de/~grinberg/algebra/gzshuf2.pdf}} \\
\textbf{project: \red \url{https://github.com/darijgr/gzshuf}}
}

% \item \href{http://arxiv.org/abs/1409.8356}{\red Darij Grinberg, Victor Reiner, \textit{Hopf Algebras in Combinatorics}, arXiv:1409.8356}.
% \item M. Hazewinkel, N. Gubareni, V.V. Kirichenko, \textit{Algebras, rings and modules. Lie algebras and Hopf algebras}, AMS 2010.

\begin{frame}
\fti{Section 1}
\begin{center}
{\LARGE \bf Section 1} \\
\noindent\rule[0.5ex]{\linewidth}{1pt}
{\Large \bf Shuffle-compatibility}
\end{center}
\vspace{1cm}
Reference:
\begin{itemize}
\item \href{https://arxiv.org/abs/1706.00750}{\red Ira M. Gessel, Yan Zhuang, \textit{Shuffle-compatible permutation statistics}, arXiv:1706.00750}.
\item See also the {\red \href{http://www.cip.ifi.lmu.de/~grinberg/algebra/urbana18a.pdf}{previous talk}} for a combinatorial introduction.
\end{itemize}
\end{frame}

\begin{frame}
\fti{Permutations \& permutation statistics: Definitions 1}

\begin{itemize}

\item This project spins off from a paper by Ira Gessel and Yan Zhuang
({\red \arxiv{1706.00750}}). \\
We prove a conjecture (shuffle-compatibility of $\Epk$)
and study a stronger version of shuffle-compatibility.

\pause

\item Let $\NN = \set{0, 1, 2, \ldots}$.

\item For $n \in \NN$, an \defn{$n$-permutation} means a tuple of $n$
distinct positive integers. \\
Example: $\tup{3, 1, 7}$ is a $3$-permutation, but
         $\tup{2, 1, 2}$ is not.

\pause

\item A \defn{permutation} means an $n$-permutation for some $n$. \\
      (\textbf{Caveat lector:} Not the usual meaning of ``permutation''.) \\
      \pause
      If $\pi$ is an $n$-permutation, then \defn{$\abs{\pi} := n$}. \\ \pause
      We say that $\pi$ is \defn{nonempty} if $n > 0$.

\pause

\item If $\pi$ is an $n$-permutation and $i \in \set{1, 2, \ldots, n}$,
      then \defn{$\pi_i$} denotes the $i$-th entry of $\pi$.

\end{itemize}
\end{frame}

\begin{frame}
\fti{Permutations \& permutation statistics: Definitions 2}

\begin{itemize}

\item Two $n$-permutations $\alpha$ and $\beta$ (with the same $n$)
      are \defn{order-equivalent} if all $i, j \in \set{1, 2, \ldots, n}$
      satisfy $\tup{\alpha_i < \alpha_j} \Longleftrightarrow
      \tup{\beta_i < \beta_j}$.

\item Order-equivalence is an equivalence relation on permutations.
      Its equivalence classes are called \defn{order-equivalence
      classes}.

\pause

\item A \defn{permutation statistic} (henceforth just \defn{statistic})
      is a map $\st$ from the set
      of all permutations (to anywhere) that is constant on each
      order-equivalence class. \\
      \textbf{Intuition:} A statistic computes some ``fingerprint''
      of a permutation that only depends on the relative order of its
      letters. \pause \\
      \textbf{Note:} The values of a statistic can be anything
      (integers, sets, etc.).

\end{itemize}
\end{frame}

\begin{frame}
\fti{Examples of permutation statistics, 1: descents et al}

\begin{itemize}

\item If $\pi$ is an $n$-permutation, then a \defn{descent} of $\pi$
      means an $i \in \set{1, 2, \ldots, n-1}$ such that
      $\pi_i > \pi_{i+1}$.

\item The \defn{descent set $\Des \pi$} of a permutation $\pi$ is
      the set of all descents of $\pi$. \\
      Thus, \defn{$\Des$} is a statistic. \\
      \textbf{Example:} $\Des \tup{3, 1, 5, 2, 4} = \set{1, 3}$.

\pause

\item The \defn{descent number $\des \pi$} of a permutation $\pi$
      is the number of all descents of $\pi$: that is,
      $\des \pi = \abs{\Des \pi}$. \\
      Thus, \defn{$\des$} is a statistic. \\
      \textbf{Example:} $\des \tup{3, 1, 5, 2, 4} = 2$.

\pause

\item The \defn{major index $\maj \pi$} of a permutation $\pi$
      is the \textbf{sum} of all descents of $\pi$. \\
      Thus, \defn{$\maj$} is a statistic. \\
      \textbf{Example:} $\maj \tup{3, 1, 5, 2, 4} = 4$.

\pause

\item The \defn{Coxeter length} $\operatorname{inv}$
      (i.e., \defn{number of inversions})
      and the \defn{set of inversions} are statistics, too.

\end{itemize}
\end{frame}

\begin{frame}
\fti{Examples of permutation statistics, 2: peaks}

\begin{itemize}

\item If $\pi$ is an $n$-permutation, then a \defn{peak} of $\pi$
      means an $i \in \set{2, 3, \ldots, n-1}$ such that
      $\pi_{i-1} < \pi_i > \pi_{i+1}$. \\
      (Thus, peaks can only exist if $n \geq 3$. \\
      The name refers to the plot of $\pi$, where peaks are
      local maxima.)

\item The \defn{peak set $\Pk \pi$} of a permutation $\pi$ is
      the set of all peaks of $\pi$. \\
      Thus, \defn{$\Pk$} is a statistic. \\
      \textbf{Examples:}
      \begin{itemize}
      \item $\Pk \tup{3, 1, 5, 2, 4} = \set{3}$.
      \item $\Pk \tup{1, 3, 2, 5, 4, 6} = \set{2, 4}$.
      \item $\Pk \tup{3, 2} = \set{}$.
      \end{itemize}

\pause

\item The \defn{peak number $\pk \pi$} of a permutation $\pi$
      is the number of all peaks of $\pi$: that is,
      $\pk \pi = \abs{\Pk \pi}$. \\
      Thus, \defn{$\pk$} is a statistic. \\
      \textbf{Example:} $\pk \tup{3, 1, 5, 2, 4} = 1$.

\end{itemize}
\vspace{10cm}
\end{frame}

\begin{frame}
\fti{Examples of permutation statistics, 3: left peaks}

\begin{itemize}

\item If $\pi$ is an $n$-permutation, then a \defn{left peak} of $\pi$
      means an $i \in \set{1, 2, \ldots, n-1}$ such that
      $\pi_{i-1} < \pi_i > \pi_{i+1}$, where we set \defn{$\pi_0 = 0$}. \\
      (Thus, left peaks are the same as peaks, except that $1$ counts
      as a left peak if $\pi_1 > \pi_2$.)

\item The \defn{left peak set $\Lpk \pi$} of a permutation $\pi$ is
      the set of all left peaks of $\pi$. \\
      Thus, \defn{$\Lpk$} is a statistic. \\
      \textbf{Examples:}
      \begin{itemize}
      \item $\Lpk \tup{3, 1, 5, 2, 4} = \set{1, 3}$.
      \item $\Lpk \tup{1, 3, 2, 5, 4, 6} = \set{2, 4}$.
      \item $\Lpk \tup{3, 2} = \set{1}$.
      \end{itemize}

\item The \defn{left peak number $\lpk \pi$} of a permutation $\pi$
      is the number of all left peaks of $\pi$: that is,
      $\lpk \pi = \abs{\Lpk \pi}$. \\
      Thus, \defn{$\lpk$} is a statistic. \\
      \textbf{Example:} $\lpk \tup{3, 1, 5, 2, 4} = 2$.

\end{itemize}
\vspace{10cm}
\end{frame}

\begin{frame}
\fti{Examples of permutation statistics, 4: right peaks}

\begin{itemize}

\item If $\pi$ is an $n$-permutation, then a \defn{right peak} of $\pi$
      means an $i \in \set{2, 3, \ldots, n}$ such that
      $\pi_{i-1} < \pi_i > \pi_{i+1}$, where we set \defn{$\pi_{n+1} = 0$}. \\
      (Thus, right peaks are the same as peaks, except that $n$ counts
      as a right peak if $\pi_{n-1} < \pi_n$.)

\item The \defn{right peak set $\Rpk \pi$} of a permutation $\pi$ is
      the set of all right peaks of $\pi$. \\
      Thus, \defn{$\Rpk$} is a statistic. \\
      \textbf{Examples:}
      \begin{itemize}
      \item $\Rpk \tup{3, 1, 5, 2, 4} = \set{3, 5}$.
      \item $\Rpk \tup{1, 3, 2, 5, 4, 6} = \set{2, 4, 6}$.
      \item $\Rpk \tup{3, 2} = \set{}$.
      \end{itemize}

\item The \defn{right peak number $\rpk \pi$} of a permutation $\pi$
      is the number of all right peaks of $\pi$: that is,
      $\rpk \pi = \abs{\Rpk \pi}$. \\
      Thus, \defn{$\rpk$} is a statistic. \\
      \textbf{Example:} $\rpk \tup{3, 1, 5, 2, 4} = 2$.

\end{itemize}
\vspace{10cm}
\end{frame}

\begin{frame}
\fti{Examples of permutation statistics, 5: exterior peaks}

\begin{itemize}

\item If $\pi$ is an $n$-permutation, then an \defn{exterior peak} of $\pi$
      means an $i \in \set{1, 2, \ldots, n}$ such that
      $\pi_{i-1} < \pi_i > \pi_{i+1}$, where we set \defn{$\pi_0 = 0$ and $\pi_{n+1} = 0$}. \\
      (Thus, exterior peaks are the same as peaks, except that
      $1$ counts if $\pi_1 > \pi_2$, and
      $n$ counts if $\pi_{n-1} < \pi_n$.)

\item The \defn{exterior peak set $\Epk \pi$} of a permutation $\pi$ is
      the set of all exterior peaks of $\pi$. \\
      Thus, \defn{$\Epk$} is a statistic. \\
      \textbf{Examples:}
      \begin{itemize}
      \item $\Epk \tup{3, 1, 5, 2, 4} = \set{1, 3, 5}$.
      \item $\Epk \tup{1, 3, 2, 5, 4, 6} = \set{2, 4, 6}$.
      \item $\Epk \tup{3, 2} = \set{1}$.
      \end{itemize}

\item Thus, $\Epk \pi = \Lpk \pi \cup \Rpk \pi$ if $n \geq 2$.

\item The \defn{exterior peak number $\epk \pi$} of a permutation $\pi$
      is the number of all exterior peaks of $\pi$: that is,
      $\epk \pi = \abs{\Epk \pi}$. \\
      Thus, \defn{$\epk$} is a statistic. \\
      \textbf{Example:} $\epk \tup{3, 1, 5, 2, 4} = 3$.

\end{itemize}
\vspace{10cm}
\end{frame}

\begin{frame}
\fti{Shuffles of permutations}

\begin{itemize}

\item Let $\pi$ and $\sigma$ be two permutations.

\item We say that $\pi$ and $\sigma$ are \defn{disjoint} if they have
      no letter in common.

\pause

\item Assume that $\pi$ and $\sigma$ are disjoint. Set
      $m = \abs{\pi}$ and $n = \abs{\sigma}$.
      An $\tup{m+n}$-permutation $\tau$ is called a \defn{shuffle} of
      $\pi$ and $\sigma$ if both $\pi$ and $\sigma$ appear as
      subsequences of $\tau$. \\
      (And thus, no other letters can appear in $\tau$.)

\item We let $S \tup{\pi, \sigma}$ be the set of all shuffles
      of $\pi$ and $\sigma$.

\item \textbf{Example:}
      \begin{align*}
      S \tup{ {\red \tup{4, 1}} , {\blue \tup{2, 5}} }
      &= \{ \tup{ {\red 4}, {\red 1}, {\blue 2}, {\blue 5} } ,
            \tup{ {\red 4}, {\blue 2}, {\red 1}, {\blue 5} } ,
            \tup{ {\red 4}, {\blue 2}, {\blue 5}, {\red 1} } , \\
      & \qquad
            \tup{ {\blue 2}, {\red 4}, {\red 1}, {\blue 5} } ,
            \tup{ {\blue 2}, {\red 4}, {\blue 5}, {\red 1} } ,
            \tup{ {\blue 2}, {\blue 5}, {\red 4}, {\red 1} }  \} .
      \end{align*}

\pause

\item Observe that $\pi$ and $\sigma$ have $\dbinom{m+n}{m}$
      shuffles, in bijection with $m$-element subsets of
      $\set{1, 2, \ldots, m+n}$.

\end{itemize}
\end{frame}

\begin{frame}
\fti{Shuffle-compatible statistics: definition}

\begin{itemize}

\item A statistic $\st$ is said to be \defn{shuffle-compatible}
      if for any two disjoint permutations $\pi$ and $\sigma$, the
      multiset
      \[
      \set{ \st\tau \mid \tau\in S\tup{\pi, \sigma} }_{\text{multiset}}
      \]
      depends only on $\st \pi$, $\st \sigma$, $\abs{\pi}$ and
      $\abs{\sigma}$.

\pause

\item In other words, $\st$ is shuffle-compatible if and only
      the distribution of
      $\st$ on the set $S \tup{ \pi , \sigma }$
      stays unchaged if $\pi$ and $\sigma$ are replaced by two
      other disjoint permutations of the same size and same $\st$-values.
      \pause \\
      In particular, it has to stay unchanged if $\pi$ and $\sigma$
      are replaced by two permutations order-equivalent to them:
      e.g., $\st$ must have the same distribution
      on the three sets
      \[
      S \tup{ {\red \tup{4, 1}} , {\blue \tup{2, 5}} },
      \qquad S \tup{ {\red \tup{2, 1}} , {\blue \tup{3, 5}} } ,
      \qquad S \tup{ {\red \tup{9, 8}} , {\blue \tup{2, 3}} } .
      \]

\end{itemize}

\end{frame}

\begin{frame}
\fti{Shuffle-compatible statistics: results of Gessel and Zhuang}

\begin{itemize}

\item Gessel and Zhuang, in {\red \arxiv{1706.00750}}, prove that
      various important statistics are shuffle-compatible (but some
      are not).

\pause

\item Statistics they show to be \textbf{shuffle-compatible}: $\Des$, $\des$,
      $\maj$, $\Pk$, $\Lpk$, $\Rpk$, $\lpk$, $\rpk$, $\epk$,
      and various others.

\pause

\item Statistics that are \textbf{not shuffle-compatible}:
      $\operatorname{inv}$, $\des + \maj$, $\maj_2$ (sending $\pi$
      to the sum of the squares of its descents),
      $\tup{\Pk, \des}$ (sending $\pi$ to $\tup{\Pk\pi, \des\pi}$),
      and others.

\pause 

\item Their proofs use a mixture of enumerative combinatorics
      (including some known formulas of MacMahon, Stanley, ...),
      quasisymmetric functions, Hopf algebra theory,
      P-partitions (and variants by Stembridge and Petersen),
      Eulerian polynomials (based on earlier work by Zhuang,
      and even earlier work by Foata and Strehl).

\pause

\item The shuffle-compatibility of $\Epk$ is left unproven
      in Gessel/Zhuang. Proving this is our first goal.

\end{itemize}

\end{frame}

\begin{frame}
\fti{Left- and right-shuffle-compatibility}

\begin{itemize}

\item We further begin the study of a finer version of
      shuffle-compatibility: ``left- and right-shuffle-compatibility''.

\item Given two disjoint nonempty permutations $\pi$ and $\sigma$,
      \begin{itemize}
      \item
      a \defn{left shuffle} of $\pi$ and $\sigma$ is a shuffle
      of $\pi$ and $\sigma$ that starts with a letter of $\pi$;
      \item
      a \defn{right shuffle} of $\pi$ and $\sigma$ is a shuffle
      of $\pi$ and $\sigma$ that starts with a letter of $\sigma$.
      \end{itemize}

\item We let $S_\prec \tup{\pi, \sigma}$ be the set of all left shuffles of $\pi$
      and $\sigma$. \\
      We let $S_\succ \tup{\pi, \sigma}$ be the set of all right shuffles of $\pi$
      and $\sigma$.

\pause

\item \only<2>{
      A statistic $\st$ is said to be \defn{left-shuffle-compatible}
      if for any two disjoint nonempty permutations $\pi$ and $\sigma$
      such that
      \[
      \text{the first entry of }\pi\text{ is greater than the first entry of }%
      \sigma,
      \]
      the multiset
      \[
      \set{ \st\tau \mid \tau\in S_\prec \tup{\pi, \sigma} }_{\text{multiset}}
      \]
      depends only on $\st \pi$, $\st \sigma$, $\abs{\pi}$ and
      $\abs{\sigma}$.
      }
      \only<3-4>{
      A statistic $\st$ is said to be \defn{right-shuffle-compatible}
      if for any two disjoint nonempty permutations $\pi$ and $\sigma$
      such that
      \[
      \text{the first entry of }\pi\text{ is greater than the first entry of }%
      \sigma,
      \]
      the multiset
      \[
      \set{ \st\tau \mid \tau\in S_\succ \tup{\pi, \sigma} }_{\text{multiset}}
      \]
      depends only on $\st \pi$, $\st \sigma$, $\abs{\pi}$ and
      $\abs{\sigma}$.
      }
      
\pause\pause

\item We'll show that $\Des$, $\des$, $\Lpk$ and $\Epk$ are
      left- and right-shuffle-compatible. \\
      (But not $\maj$ or $\Rpk$.)

\end{itemize}
\vspace{10cm}
\end{frame}

\begin{frame}
\fti{Section 2}
\begin{center}
{\LARGE \bf Section 2} \\
\noindent\rule[0.5ex]{\linewidth}{1pt}
{\Large \bf The algebraic approach: $\QSym$ and kernels}
\end{center}
\vspace{1cm}
Reference:
\begin{itemize}
\item \href{https://arxiv.org/abs/1706.00750}{\red Ira M. Gessel, Yan Zhuang, \textit{Shuffle-compatible permutation statistics}, arXiv:1706.00750}.
\item \href{http://arxiv.org/abs/1409.8356}{\red Darij Grinberg, Victor Reiner, \textit{Hopf Algebras in Combinatorics}, arXiv:1409.8356},
      and various other texts on combinatorial Hopf algebras.
\end{itemize}
\end{frame}

\begin{frame}
\fti{Descent statistics}

\begin{itemize}

\item Gessel and Zhuang prove \textbf{most} of their shuffle-compatibilities
      algebraically. Their methods involve combinatorial Hopf
      algebras ($\QSym$ and $\NSym$).

\item These methods work for \textbf{descent statistics} only.
      What is a descent statistic?

\pause

\item A \defn{descent statistic} is a statistic $\st$ such that
      $\st \pi$ depends only on $\abs{\pi}$ and $\Des\pi$
      (in other words: if $\pi$ and $\sigma$ are two
      $n$-permutations with $\Des\pi = \Des\sigma$, then
      $\st\pi = \st\sigma$). \\
      \textbf{Intuition:} A descent statistic is a statistic
      which ``factors through $\Des$ in each size''.

\end{itemize}
\end{frame}

\begin{frame}
\fti{Compositions: definitions}

\begin{itemize}
      
\item A \defn{composition} is a finite list of positive integers.
      \\
      A \defn{composition of $n \in \NN$} is a composition whose
      entries sum to $n$. \pause

\item For example, the compositions of $5$ are
      \begin{align*}
      &
      \tup{1, 1, 1, 1, 1}, \qquad
      \tup{1, 1, 1, 2}, \qquad
      \tup{1, 1, 2, 1}, \qquad
      \tup{1, 1, 3}, \\ &
      \tup{1, 2, 1, 1}, \qquad
      \tup{1, 2, 2}, \qquad
      \tup{1, 3, 1}, \qquad
      \tup{1, 4}, \\ &
      \tup{2, 1, 1, 1}, \qquad
      \tup{2, 1, 2}, \qquad
      \tup{2, 2, 1}, \qquad
      \tup{2, 3}, \\ &
      \tup{3, 1, 1}, \qquad
      \tup{3, 2}, \qquad
      \tup{4, 1}, \qquad
      \tup{5} .
      \end{align*}

\pause

\item The \defn{size $\abs{\alpha}$} of a composition $\alpha$
      is defined by $\abs{\alpha} := \alpha_1 + \alpha_2 + \cdots + \alpha_k$.
      \\
      Thus, a composition of $n$ is the same as a composition
      of size $n$.

\pause

\item For each positive integer $n$, there are exactly $2^{n-1}$
      compositions of $n$. Why?

\end{itemize}

\end{frame}

\begin{frame}
\fti{Compositions vs. subsets: The $\Des$ and $\Comp$ bijections}

\begin{itemize}

\item For each $k \in \NN$, set $\ive{k} = \set{1, 2, \ldots, k}$.

\pause

\item Let $n$ be a positive integer. \\
      Then, there are mutually inverse bijections
      \begin{align*}
                \set{\text{compositions of } n}
               &\overset{\Des}{\underset{\Comp}{\rightleftarrows}}
               \set{\text{subsets of } \ive{n-1} }, \\
               \tup{i_1, i_2, \ldots, i_k}
               &\mapsto \set{i_1 + i_2 + \cdots + i_j \mid j \in \ive{k-1} } , \\
               \hspace{-0.8cm}\tup{s_1 - s_0, s_2 - s_1, \ldots, s_{k+1} - s_k }
               &\mapsfrom \set{s_1 < s_2 < \cdots < s_k}
      \end{align*}
      (using the notations $s_0 = 0$ and $s_{k+1} = n$).
      
      \only<2>{
      \textbf{Caveat lector:}
      \begin{align*}
      \Des\tup{\tup{1, 5, 2} \text{ the composition}} &= \set{1, 6} ; \\
      \Des\tup{\tup{1, 5, 2} \text{ the permutation}} &= \set{2} .
      \end{align*}
      Context must disambiguate.
      }
      
\pause
      
\item If $\pi$ is an $n$-permutation, then $\Comp\tup{\Des \pi}$ is
      called the \defn{descent composition} of $\pi$, and is written
      \defn{$\Comp\pi$}.

\pause

\only<4>{
\item Thus, a descent statistic is a statistic $\st$ that factors through
      $\Comp$ (that is, $\st\pi$ depends only on $\Comp\pi$).
}

\pause

\item If $\st$ is a descent statistic, then we use the notation
      \defn{$\st \alpha$} (where $\alpha$ is a composition) for $\st \pi$,
      where $\pi$ is any permutation with $\Comp \pi = \alpha$. \\
      (Again, this notation is ambiguous if compositions are not
      distinguished from permutations.)

\end{itemize}

\vspace{10cm}

\end{frame}

\begin{frame}
\fti{Descent statistics: examples}

\begin{itemize}

\item Almost all of our statistics so far are descent statistics.
     Examples:

\pause

\item $\Des$, $\des$ and $\maj$ are descent statistics.

\pause

\item $\Pk$ is a descent statistic: If $\pi$ is an $n$-permutation,
      then
      \[
      \Pk \pi = \tup{\Des \pi} \setminus \tup{\tup{\Des \pi \cup \set{0}} + 1} ,
      \]
      where for any set $K$ of integers and any integer $a$
      we set \defn{$K + a = \set{k + a \mid k \in K}$}.

\item Similarly, $\Lpk$, $\Rpk$ and $\Epk$ are descent statistics.

\pause

\item $\inv$ is not a descent statistic:
      The permutations $\tup{2,1,3}$ and $\tup{3,1,2}$ have the same
      descents, but different numbers of inversions.

\pause

\item \textbf{Question (Gessel \& Zhuang).}
      Is every shuffle-compatible statistic a descent statistic?

\end{itemize}

\end{frame}

\begin{frame}
\fti{Power series \& symmetric functions}

\begin{itemize}

\item Let's now talk about power series, which are crucial to the
      algebraic approach to shuffle-compatibility.

\item Consider the ring $\QQ\left[\left[x_1,x_2,x_3,\ldots\right]\right]$
      of formal power series in countably many indeterminates.

\pause

\item A formal power series $f$ is said to be \defn{bounded-degree} if
      the monomials it contains are bounded (from above) in degree.

\pause

\item A formal power series $f$ is said to be \defn{symmetric} if it
      is invariant under permutations of the indeterminates. \\
      Equivalently, if its coefficients in front of
      $x_{i_1}^{a_1} x_{i_2}^{a_2} \cdots x_{i_k}^{a_k}$ and
      $x_{j_1}^{a_1} x_{j_2}^{a_2} \cdots x_{j_k}^{a_k}$
      are equal whenever $i_1, i_2, \ldots, i_k$ are distinct and
      $j_1, j_2, \ldots, j_k$ are distinct.

\only<1-3>{\item For example:
\begin{itemize}
\item $1 + x_1 + x_2^3$ is bounded-degree but not symmetric.
\item $\left(1+x_1\right)\left(1+x_2\right)\left(1+x_3\right) \cdots $ is symmetric but not bounded-degree.
\end{itemize}}
\only<4>{\item The symmetric bounded-degree power series form a
$\QQ$-subalgebra
$\Sym$ of $\QQ\left[\left[x_1,x_2,x_3,\ldots\right]\right]$,
called the \defn{ring of symmetric functions} over $\QQ$
(often denoted by $\Lambda$).
This talk is not about it.}

\vspace{3cm}
\end{itemize}

\end{frame}

\begin{frame}
\fti{Quasisymmetric functions, part 1: definition}

\begin{itemize}

\item We shall now define the quasisymmetric functions -- a bigger algebra than $\Sym$, but still with many of its nice properties.

\item A formal power series $f$ (still in $\QQ\left[\left[x_1,x_2,x_3,\ldots\right]\right]$) is said to be \defn{quasisymmetric} if its coefficients in front of $x_{i_1}^{a_1} x_{i_2}^{a_2} \cdots x_{i_k}^{a_k}$ and $x_{j_1}^{a_1} x_{j_2}^{a_2} \cdots x_{j_k}^{a_k}$ are equal whenever $i_1 < i_2 < \cdots < i_k$ and $j_1 < j_2 < \cdots < j_k$.

\item For example:
\begin{itemize}
\item Every symmetric power series is quasisymmetric.

\item $\sum\limits_{i<j} x_i^2 x_j = x_1^2 x_2 + x_1^2 x_3 + x_2^2 x_3 + x_1^2 x_4 + \cdots$ is quasisymmetric, but not symmetric.

\end{itemize}

\pause

\item Let \defn{$\QSym$} be the set of all quasisymmetric bounded-degree power series in $\QQ\left[\left[x_1,x_2,x_3,\ldots\right]\right]$. This is a $\QQ$-subalgebra, called the \defn{ring of quasisymmetric functions} over $\QQ$. (Gessel, 1980s.) \pause

\only<3>{
\item We have $\Sym \subseteq \QSym \subseteq \QQ\left[\left[x_1,x_2,x_3,\ldots\right]\right]$.
}
\pause

\item The $\QQ$-vector space $\QSym$ has several combinatorial bases.
      We will use two of them: the monomial basis and the
      fundamental basis.

\end{itemize}

\end{frame}

\begin{frame}
\fti{Quasisymmetric functions, part 2: the monomial basis}

\begin{itemize}

\item For every composition
      $\alpha = \left(\alpha_1, \alpha_2, \ldots, \alpha_k\right)$, define
\begin{align*}
M_\alpha &= \sum\limits_{i_1 < i_2 < \cdots < i_k} x_{i_1}^{\alpha_1} x_{i_2}^{\alpha_2} \cdots x_{i_k}^{\alpha_k} \\
&= \text{sum of all monomials whose nonzero exponents } \\
& \qquad \text{are } \alpha_1, \alpha_2, \ldots, \alpha_k \text{ in \textbf{this} order}.
\end{align*}
This is a homogeneous power series of degree $\abs{\alpha}$. % (the size of $\alpha$, defined by $\abs{\alpha} := \alpha_1 + \alpha_2 + \cdots + \alpha_k$).

\only<1>{\item Examples:
\begin{itemize}
\item $M_{\left(\right)} = 1$.
\item $M_{\left(1,1\right)} = \sum\limits_{i<j} x_i x_j = x_1 x_2 + x_1 x_3 + x_2 x_3 + x_1 x_4 + x_2 x_4 + \cdots$.
\item $M_{\left(2,1\right)} = \sum\limits_{i<j} x_i^2 x_j = x_1^2 x_2 + x_1^2 x_3 + x_2^2 x_3 + \cdots$.
\item $M_{\left(3\right)} = \sum\limits_i x_i^3 = x_1^3 + x_2^3 + x_3^3 + \cdots$.
\end{itemize}
%Note: $m_{\left(2,1\right)} = M_{\left(2,1\right)} + M_{\left(1,2\right)}$.
}
\only<2>{
\item The family $\left(M_\alpha\right)_{\alpha \text{ is a composition}}$ is a basis of the $\QQ$-vector space $\QSym$, called the \defn{monomial basis} (or $M$-basis).
}

\end{itemize}

\vspace{4cm}

\end{frame}

\begin{frame}
\fti{Quasisymmetric functions, part 3: the fundamental basis}

\begin{itemize}

\item For every composition
      $\alpha = \left(\alpha_1, \alpha_2, \ldots, \alpha_k\right)$, define
\begin{align*}
F_\alpha &= \sum\limits_{\substack{i_1 \leq i_2 \leq \cdots \leq i_n;\\ i_j < i_{j+1} \text{ for all } j \in \Des\alpha}} x_{i_1} x_{i_2} \cdots x_{i_n} \\
&= \sum\limits_{\substack{\beta \text{ is a composition of } n; \\ \Des \beta \supseteq \Des \alpha}} M_\beta , \qquad \text{where } n = \abs{\alpha} .
\end{align*}
This is a homogeneous power series of degree $\abs{\alpha}$ again.

\only<1>{\item Examples:
\begin{itemize}
\item $F_{\left(\right)} = 1$.
\item $F_{\left(1,1\right)} = \sum\limits_{i<j} x_i x_j = x_1 x_2 + x_1 x_3 + x_2 x_3 + x_1 x_4 + x_2 x_4 + \cdots$.
\item $F_{\left(2,1\right)} = \sum\limits_{i \leq j < k} x_i x_j x_k$.
\item $F_{\left(3\right)} = \sum\limits_{i \leq j \leq k} x_i x_j x_k$.
\end{itemize}
}
\only<2>{
\item The family $\left(F_\alpha\right)_{\alpha \text{ is a composition}}$ is a basis of the $\QQ$-vector space $\QSym$, called the \defn{fundamental basis} (or $F$-basis). \\
Sometimes, $F_\alpha$ is also denoted $L_\alpha$.
}

\end{itemize}

\vspace{4cm}

\end{frame}

\begin{frame}
\fti{The product formula for the $F_\alpha$}

\begin{itemize}

\item What connects $\QSym$ with shuffles of permutations is the following fact:
      \\
      \textbf{Theorem.} If $\pi$ and $\sigma$ are two disjoint permutations,
      then
      \[
      F_{\Comp \pi} \cdot F_{\Comp \sigma}
      = \sum\limits_{\tau \in S\tup{\pi, \sigma}} F_{\Comp \tau} .
      \]

\pause

\item This theorem yields that $\Des$ is shuffle-compatible. Why?
      \pause
      \begin{itemize}
      \item
      \only<3>{
      Let $\pi, \pi', \sigma, \sigma'$ be permutations with
      $\abs{\pi} = \abs{\pi'}$ and $\abs{\sigma} = \abs{\sigma'}$
      and $\Des\pi = \Des\pi'$ and $\Des\sigma = \Des\sigma'$. \\
      We must prove that
      \begin{align*}
      &\set{ \Des\tau \mid \tau\in S\tup{\pi, \sigma} }_{\text{multiset}} \\
      =
      &\set{ \Des\tau \mid \tau\in S\tup{\pi', \sigma'} }_{\text{multiset}} .
      \end{align*}
      }
      \only<4-7>{
      Let $\pi, \pi', \sigma, \sigma'$ be permutations with
      $\Comp\pi = \Comp\pi'$ and $\Comp\sigma = \Comp\sigma'$. \\
      We must prove that
      }
      \only<4>{
      \begin{align*}
      &\set{ \Comp\tau \mid \tau\in S\tup{\pi, \sigma} }_{\text{multiset}} \\
      =
      &\set{ \Comp\tau \mid \tau\in S\tup{\pi', \sigma'} }_{\text{multiset}}
      \end{align*}
      (this is equivalent to what we just said,
      since $\Comp \pi$ encodes the same
      data as $\Des \pi$ and $\abs{\pi}$ together).
      }
      \only<5>{
      \[
      \sum\limits_{\tau \in S\tup{\pi, \sigma}} F_{\Comp \tau}
      = \sum\limits_{\tau \in S\tup{\pi', \sigma'}} F_{\Comp \tau} 
      \]
      (this is equivalent to what we just said, since the
      $F_\alpha$ for $\alpha$ ranging over all compositions
      are linearly independent).
      }
      \only<6-7>{
      \[
      F_{\Comp \pi} \cdot F_{\Comp \sigma}
      = F_{\Comp \pi'} \cdot F_{\Comp \sigma'}
      \]
      (this is equivalent to what we just said, by the Theorem
      above).
      }
      \only<7>{
      \\ But this follows from assumptions.
      }
      \end{itemize}

\end{itemize}
\vspace{10cm}
\end{frame}

\begin{frame}
\fti{Shuffle-compatibility of $\des$}

\begin{itemize}

\only<1>{
\item The same technique works for some other statistics.
      For example, we can show that $\des$ is shuffle-compatible.
}

\pause

\item For any $n \in \NN$ and $k \in \NN$, define the polynomial
      \[
      f_{n, k} = x^n \dbinom{p - k + n}{n} \in \QQ\ive{p, x}.
      \]

\pause

\item \textbf{Corollary (of preceding Theorem).}
      If $\pi$ and $\sigma$ are two disjoint permutations,
      with $n = \abs{\pi}$ and $m = \abs{\sigma}$,
      then
      \[
      f_{n, \des \pi} \cdot f_{m, \des \sigma}
      = \sum\limits_{\tau \in S\tup{\pi, \sigma}} f_{n+m, \des \tau} .
      \]

\pause

\only<3-4>{
\item \textbf{Proof idea (from Gessel/Zhuang).}
      There is a $\QQ$-algebra homomorphism $\QSym \to \QQ\ive{p, x}$
      sending each $g \in \QSym$ to
      $g \tup{ \underbrace{x, x, \ldots, x}_{p \text{ times}} ,
                       0, 0, 0, \ldots }$
      (yes, this can be made sense of).
      \only<4>{
      This is a variant of the \defn{(generic) principal specialization}.
      }
      \pause
      \\ Check that it sends $F_{\Comp \pi}$ to $f_{n, \des \pi}$
      for any $n$-permutation $\pi$.
      Then, apply it to the preceding Theorem.
}

\only<5-8>{
\item This corollary yields that $\des$ is shuffle-compatible. Why?
      \pause
      \begin{itemize}
      \item
      Let $\pi, \pi', \sigma, \sigma'$ be permutations with
      $\abs{\pi} = \abs{\pi'}$ and $\abs{\sigma} = \abs{\sigma'}$
      and $\des\pi = \des\pi'$ and $\des\sigma = \des\sigma'$. \\
      We must prove that
      \only<5>{
      \begin{align*}
      &\set{ \des\tau \mid \tau\in S\tup{\pi, \sigma} }_{\text{multiset}} \\
      =
      &\set{ \des\tau \mid \tau\in S\tup{\pi', \sigma'} }_{\text{multiset}} .
      \end{align*}
      }
      \only<6>{
      \[
      \sum\limits_{\tau \in S\tup{\pi, \sigma}} f_{n+m, \des \tau}
      = \sum\limits_{\tau \in S\tup{\pi', \sigma'}} f_{n+m, \des \tau} ,
      \]
      where $n = \abs{\pi} = \abs{\pi'}$ and $m = \abs{\sigma} = \abs{\sigma'}$
      (this is equivalent to what we just said, since the
      $f_{n, k}$ for $n, k \in \NN$
      are linearly independent).
      }
      \only<7-8>{
      \[
      f_{n, \des \pi} \cdot f_{m, \des \sigma}
      = f_{n, \des \pi'} \cdot f_{m, \des \sigma'}
      \]
      (this is equivalent to what we just said, by the Corollary
      above).
      }
      \only<8>{
      \\ But this follows from assumptions.
      }
      \end{itemize}
}

\end{itemize}
\vspace{10cm}
\end{frame}

\begin{frame}
\fti{The kernel criterion for shuffle-compatibility, 1}

\begin{itemize}

\item The above arguments can be abstracted into a general
      criterion for shuffle-compatibility of a descent statistic
      (Gessel and Zhuang, in {\red \arxiv{1706.00750v2}},
      Section 4.1).
      \\
      $\QSym$ and $\QQ\ive{p, x}$ get replaced by a ``shuffle
      algebra'' with an algebra homomorphism from $\QSym$.

\item We shall give our own variant of the criterion.

\end{itemize}

\end{frame}

\begin{frame}
\fti{The kernel criterion for shuffle-compatibility, 2}

\begin{itemize}

\item If $\st$ is a descent statistic, then two compositions
      $\alpha$ and $\beta$ are said to be \defn{$\st$-equivalent}
      if $\abs{\alpha} = \abs{\beta}$ and $\st\alpha = \st\beta$.
      \\ (Remember: $\st\alpha$ means $\st\pi$ for any permutation
      $\pi$ satisfying $\Comp\pi = \alpha$.)

\pause

\item The \defn{kernel $\calK_{\st}$} of a descent statistic $\st$
      is the $\QQ$-vector subspace of $\QSym$ spanned by all
      differences of the form $F_\alpha - F_\beta$, with $\alpha$
      and $\beta$ being two $\st$-equivalent compositions:
      \[
      \calK_{\st} = \left< F_\alpha - F_\beta \ 
                              \mid \ \abs{\alpha} = \abs{\beta} \text{ and }
                                   \st \alpha = \st \beta \right>_\QQ .
      \]

\pause

\item \textbf{Theorem.} The descent statistic $\st$ is
      shuffle-compatible if and only if $\calK_{\st}$ is an
      ideal of $\QSym$.

\end{itemize}

\end{frame}

\begin{frame}
\fti{Section 3}
\begin{center}
{\LARGE \bf Section 3} \\
\noindent\rule[0.5ex]{\linewidth}{1pt}
{\Large \bf The exterior peak set}
\end{center}
\vspace{1cm}
References:
\begin{itemize}
\item \href{https://github.com/darijgr/gzshuf}{\red Darij Grinberg, \textit{Shuffle-compatible permutation statistics II: the exterior peak set}, draft}.
\item \href{http://www.ams.org/journals/tran/1997-349-02/S0002-9947-97-01804-7/}{\red John R. Stembridge, \textit{Enriched P-partitions}, Trans. Amer. Math. Soc. 349 (1997), no. 2, pp. 763--788}.
\item \href{https://doi.org/10.1016/j.aim.2006.05.0160}{\red T. Kyle Petersen, \textit{Enriched P-partitions and peak algebras}, Adv. in Math. 209 (2007), pp. 561--610}.
\end{itemize}
\end{frame}

\begin{frame}
\fti{Roadmap to $\Epk$}

\begin{itemize}

\item We will now outline our proof that $\Epk$ is shuffle-compatible.

\item The main idea is to imitate the above proof for $\Des$,
      but instead of $F_{\Comp \pi}$ we'll now have
      some different power series (not in $\QSym$).

\pause

\item The idea is not new.
      This is how $\Pk$, $\Lpk$ and $\Rpk$ were proven
      shuffle-compatible.

\pause

\item The main tool is the concept of
      \textbf{$\mathcal{Z}$-enriched $P$-partitions}:
      a generalization of
      \begin{itemize}
      \item $P$-partitions (Stanley 1972);
      \item enriched $P$-partitions (Stembridge 1997);
      \item left enriched $P$-partitions (Petersen 2007),
      \end{itemize}
      which are used in the proofs for $\Des$, $\Pk$
      and $\Lpk$, respectively. \pause
      \\ (Yes, the $F_{\Comp \pi} \cdot F_{\Comp \sigma}$
      theorem we used in proving $\Des$ follows from the
      theory of $P$-partitions.)

\pause

\item The idea is simple, but the proof has technical
      parts I am not showing.

\end{itemize}

\end{frame}

\begin{frame}
\fti{Labeled posets}

\begin{itemize}

\item A \defn{labeled poset} means a pair $\left(  P,\gamma\right)  $ consisting
of a finite poset $P=\left(  X,\leq\right)  $ and an injective map
$\gamma:X\rightarrow A$ into some totally ordered set $A$. The injective map
$\gamma$ is called the \defn {labeling} of the labeled poset $\left(
P,\gamma\right)  $.

\end{itemize}

\end{frame}

\begin{frame}
\fti{$\calN$ and $\calZ$: definitions}

\begin{itemize}

\item Fix a totally ordered set \defn{$\mathcal{N}$}, and denote its strict order
relation by \defn{$\prec$}.

\item Let \defn{$+$} and \defn{$-$} be two distinct symbols. \\
Let \defn{$\mathcal{Z}$} be a subset of the set $\mathcal{N}\times\left\{  +,-\right\}
$.

\item \textbf{Intuition:} $\calN$ is a set of letters that will index our
      indeterminates. \\
      $\calZ$ is a set of ``signed letters'', which are pairs of a letter
      in $\calN$ and a sign in $\set{+,-}$.
      (Not all such pairs must lie in $\calZ$.)

\pause

% \item For each $q=\left(  n,s\right)  \in\mathcal{Z}$, we denote the element
% $n\in\mathcal{N}$ by \defn{$\left\vert q\right\vert $}, and we call the element
% $s\in\left\{  +,-\right\}  $ the \textit{sign} of $q$.

\item If $n\in\mathcal{N}$,
then we will denote the two elements $\left(  n,+\right)  $ and $\left(
n,-\right)  $ of $\mathcal{N}\times\left\{  +,-\right\}  $ by \defn{$+n$} and \defn{$-n$}, respectively.

\pause

\item Let us totally order the set $\mathcal{Z}$ in such a way that the (strict)
order relation $\prec$ satisfies%
\begin{align*}
\left(  n,s\right)  &\prec\left(  n^{\prime},s^{\prime}\right)  \text{ if and
only if either }n\prec n^{\prime} \\
& \qquad \qquad \text{ or }\left(  n=n^{\prime}\text{ and
}s=-\text{ and }s^{\prime}=+\right)  .
\end{align*}

\pause

\item Let $\operatorname*{Pow}\mathcal{N}$ be the ring of all power series over
$\mathbb{Q}$ in the indeterminates $x_{n}$ for $n\in\mathcal{N}$.

\end{itemize}

\end{frame}

\begin{frame}
\fti{$\calN$ and $\calZ$: example}

\begin{itemize}

\item For an example of the setting just introduced, take
$\calN = \NN$ with $\prec$ being the usual order. Then,
\[
\calZ \subseteq \NN \times \set{+, -}
= \set{ -0, +0, -1, +1, -2, +2, \ldots } .
\]
\textbf{Caveat lector:} $-0 \neq +0$, since these are shorthands for pairs, not
numbers.

\pause

\item The total order $\prec$ on $\calZ$ is the restriction of
\[
-0 \prec +0 \prec -1 \prec +1 \prec -2 \prec +2 \prec \cdots .
\]

\pause

\item $\operatorname*{Pow}\mathcal{N}
= \QQ\ive{\ive{x_0, x_1, x_2, \ldots}}$.

\end{itemize}

\end{frame}

\begin{frame}
\fti{$\calZ$-enriched $\tup{P,\gamma}$-partitions: definition}

\begin{itemize}

\item Now, let $\left(  P,\gamma\right)  $ be a labeled poset. A
\defn{$\mathcal{Z}$-enriched $\left(  P,\gamma\right)  $-partition}
means a map $f:P\rightarrow\mathcal{Z}$ such that for all $x<y$ in $P$, the
following conditions hold:

\begin{enumerate}
\item[\textbf{(i)}] We have $f\left(  x\right)  \preccurlyeq f\left(
y\right)  $.

\item[\textbf{(ii)}] If $f\left(  x\right)  =f\left(  y\right)  =+n$ for some
$n\in\mathcal{N}$, then $\gamma\left(  x\right)  <\gamma\left(  y\right)  $.

\item[\textbf{(iii)}] If $f\left(  x\right)  =f\left(  y\right)  =-n$ for some
$n\in\mathcal{N}$, then $\gamma\left(  x\right)  >\gamma\left(  y\right)  $.
\end{enumerate}

(Keep in mind: $\calN$ and $\calZ$ are fixed.)

\pause

\item \textbf{(Attempt at) intuition:}
      A $\calZ$-enriched $\tup{P,\gamma}$-partition is a map
      $f : P \to \calZ$ (that is, assigning a signed letter to each poset
      element) which
      \begin{enumerate}
      \item[\textbf{(i)}] is weakly increasing on $P$;
      \item[\textbf{(ii) + (iii)}] is occasionally strictly increasing,
                                   when $\gamma$ and the sign of the
                                   $f$-value ``are out of alignment''.
      \end{enumerate}

\end{itemize}

\end{frame}

\begin{frame}
\fti{$\calZ$-enriched $\tup{P,\gamma}$-partitions: example}

\begin{itemize}

\item Let $P$ be the poset with the following Hasse
diagram:%
\[
\xymatrix@R=1pc{
& b \are[dl] \are[dr] \\
c \are[dr] & & d \are[dl] \\
& a
}%
%EndExpansion
\]
and let $\gamma:P\rightarrow\mathbb{Z}$ be a labeling that satisfies
$\gamma\left(  a\right)  <\gamma\left(  b\right)  <\gamma\left(  c\right)
<\gamma\left(  d\right)  $ (for example, $\gamma$ could be the map that sends
$a,b,c,d$ to $2,3,5,7$, respectively). Then, a $\mathcal{Z}$-enriched $\left(
P,\gamma\right)  $-partition is a map $f:P\rightarrow\mathcal{Z}$ satisfying
the following conditions:

\begin{enumerate}
\item[\textbf{(i)}] We have $f\left(  a\right)  \preccurlyeq f\left(
c\right)  \preccurlyeq f\left(  b\right)  $ and $f\left(  a\right)
\preccurlyeq f\left(  d\right)  \preccurlyeq f\left(  b\right)  $.

\item[\textbf{(ii)}] We cannot have $f\left(  c\right)  =f\left(  b\right)
=+n$ with $n\in\mathcal{N}$. Also, we cannot have $f\left(  d\right)
=f\left(  b\right)  =+n$ with $n\in\mathcal{N}$.

\item[\textbf{(iii)}] We cannot have $f\left(  a\right)  =f\left(  c\right)
=-n$ with $n\in\mathcal{N}$. Also, we cannot have $f\left(  a\right)
=f\left(  d\right)  =-n$ with $n\in\mathcal{N}$.
\end{enumerate}

% For example, if $\mathcal{N}=\mathbb{P}$ (the totally ordered set of positive
% integers, with its usual ordering) and $\mathcal{Z}=\mathcal{N}\times\left\{
% +,-\right\}  $, then the map $f:P\rightarrow\mathcal{Z}$ sending $a,b,c,d$ to
% $+2,-3,+2,-3$ (respectively) is a $\mathcal{Z}$-enriched $\left(
% P,\gamma\right)  $-partition. Notice that the total ordering on $\mathcal{Z}$
% in this case is given by%
% \[
% -1\prec+1\prec-2\prec+2\prec-3\prec+3\prec\cdots,
% \]
% rather than by the familiar total order on $\mathbb{Z}$.

\end{itemize}

\end{frame}

\begin{frame}
\fti{$\calZ$-enriched $\tup{P,\gamma}$-partitions: revisiting the literature}

\begin{itemize}

\item Consider again the case when $\calN = \NN$ with $\prec$ being the usual order.
Let us see what $\calZ$-enriched $\tup{P,\gamma}$-partitions are,
depending on $\calZ$.

\pause

\item If $\calZ = \NN \times \set{+} = \set{+0 \prec +1 \prec +2 \prec \cdots}$,
then the $\calZ$-enriched $\tup{P,\gamma}$-partitions are just
the {\blue (usual) $\tup{P,\gamma}$-partitions} into $\NN$ (up to renaming
$n$ as $+n$).

\pause
\item If $\calZ = \NN \times \set{+,-}
= \set{-0 \prec +0 \prec -1 \prec +1 \prec -2 \prec +2 \prec \cdots}$, then the
$\calZ$-enriched $\tup{P,\gamma}$-partitions are
{\blue Stembridge's enriched $\tup{P,\gamma}$-partitions} (up to renaming
$n$ as $n-1$).

\pause
\item If $\calZ = \tup{\NN \times \set{+,-}} \setminus \set{-0}
= \set{+0 \prec -1 \prec +1 \prec -2 \prec +2 \prec \cdots}$, then the
$\calZ$-enriched $\tup{P,\gamma}$-partitions are
{\blue Petersen's left enriched $\tup{P,\gamma}$-partitions}.

\pause

\item We shall later focus on the case when
$\calN = \NN \cup \set{\infty}$ and
$\calZ = \tup{\calN \times \set{+,-}} \setminus \set{-0, +\infty}$.

\end{itemize}

\end{frame}

\begin{frame}
\fti{$\calE\tup{P,\gamma}$ and $\calL\tup{P}$}

\begin{itemize}

\item A few more notations are needed.

\item If $\left(  P,\gamma\right)  $ is a labeled poset, then
\defn{$\calE\tup{P,\gamma}$}
shall denote the set of all $\mathcal{Z}$-enriched $\left(
P,\gamma\right)  $-partitions.

\pause

\item If $P$ is any poset, then \defn{$\calL\tup{P}$} shall denote the set
of all linear extensions of $P$. \\
A linear extension of $P$ shall be understood
simultaneously as a totally ordered set extending $P$ and as a list $\left(
w_{1},w_{2},\ldots,w_{n}\right)  $ of all elements of $P$ such that no two
integers $i<j$ satisfy $w_{i}\geq w_{j}$ in $P$.

\end{itemize}

\end{frame}

\begin{frame}
\fti{Any $\calE\tup{P,\gamma}$-partition has its favorite linear extension}

\begin{itemize}

\item \textbf{Proposition.} For any labeled poset $\left(  P,\gamma\right)  $, we
have
\[
\mathcal{E}\left(  P,\gamma\right)  =\bigsqcup_{w\in\mathcal{L}\left(
P\right)  }\mathcal{E}\left(  w,\gamma\right)  .
\]

\item This is a generalization of a standard result on $P$-partitions
      (``Stanley's main lemma''), and is proven by the same reasoning.

\end{itemize}

\end{frame}

\begin{frame}
\fti{The power series $\Gamma_{\calZ}\tup{P,\gamma}$}

\begin{itemize}

\item Let $\left(  P,\gamma\right)  $ be a labeled poset. We
define a power series \defn{$\Gamma_{\mathcal{Z}}\left(  P,\gamma\right)
\in\operatorname*{Pow}\mathcal{N}$} by
\[
\Gamma_{\mathcal{Z}}\left(  P,\gamma\right)  =\sum_{f\in\mathcal{E}\left(
P,\gamma\right)  }\prod_{p\in P}x_{\left\vert f\left(  p\right)  \right\vert
}.
\]
Here, \defn{$\abs{f\tup{p}} \in \calN$} is defined to be the first entry of
$f\tup{p}$ (recall: $f\tup{p}$ is a pair of an element of $\calN$
and a sign in $\set{+,-}$).

\pause

\item This generalizes the classical quasisymmetric $P$-partition
enumerators (which give the fundamental basis $F_\alpha$ when
$P$ is totally ordered).

\pause

\only<3>{
\item \textbf{Corollary.} For any labeled poset $\left(  P,\gamma\right)  $, we have%
\[
\Gamma_{\mathcal{Z}}\left(  P,\gamma\right)  =\sum_{w\in\mathcal{L}\left(
P\right)  }\Gamma_{\mathcal{Z}}\left(  w,\gamma\right)  .
\]
}

\only<4>{
\item \textbf{Question.} Where do these $\Gamma_{\calZ}\tup{P,\gamma}$ live
(other than in $\operatorname{Pow}\calN$) ? \\
I don't know a good answer; it should be a generalization of $\QSym$.
\\
Jia Huang's work ({\red \arxiv{1506.02962v2}}) looks relevant.
}

\end{itemize}

\vspace{5cm}

\end{frame}

\begin{frame}
\fti{Disjoint unions give product of $\Gamma$'s}

\begin{itemize}

\item Let $P$ be any set. Let $A$ be a totally ordered set. Let $\gamma:P\rightarrow
A$ and $\delta:P\rightarrow A$ be two maps. We say that $\gamma$ and $\delta$
are \defn{order-equivalent} if the following holds: For every pair $\left(
p,q\right)  \in P\times P$, we have $\gamma\left(  p\right)  \leq\gamma\left(
q\right)  $ if and only if $\delta\left(  p\right)  \leq\delta\left(
q\right)  $.

\pause

\item \textbf{Proposition.} Let $\left(  P,\gamma\right)  $ and $\left(  Q,\delta
\right)  $ be two labeled posets. Let $\left(  P\sqcup Q,\varepsilon\right)  $
be the labeled poset
\begin{itemize}
\item for which $P\sqcup Q$ is the disjoint union of
$P$ and $Q$, and
\item whose labeling $\varepsilon$ is such that the restriction of
$\varepsilon$ to $P$ is order-equivalent to $\gamma$ and such that the
restriction of $\varepsilon$ to $Q$ is order-equivalent to $\delta$.
\end{itemize}
Then,%
\[
\Gamma_{\mathcal{Z}}\left(  P,\gamma\right)  \cdot
\Gamma_{\mathcal{Z}}\left(
Q,\delta\right)  =\Gamma_{\mathcal{Z}}\left(  P\sqcup Q,\varepsilon\right)  .
\]

\item Again, the proof is simple.

\end{itemize}

\end{frame}

\begin{frame}
\fti{From a permutation to a labeled poset}

\begin{itemize}

\item Let $n\in\mathbb{N}$. Write $\ive{n}$ for $\set{1,2,\ldots,n}$. \\
Let $\pi$ be any $n$-permutation.
Consider $\pi$ as an injective map $\ive{n} \to \set{1,2,3,\ldots}$
(sending $i$ to $\pi_i$).
Thus, $\left(  \left[  n\right]  ,\pi\right)$ is a labeled poset.
We define \defn{$\Gamma_{\mathcal{Z}}\left(  \pi\right)  $} to be the
power series $\Gamma_{\mathcal{Z}}\left(  \left[  n\right]  ,\pi\right)  $.

\pause

\only<2>{
\item Explicitly:
\[
\Gamma_{\calZ}\tup{\pi}
=
\sum x_{\abs{j_1}} x_{\abs{j_2}} \cdots x_{\abs{j_n}} ,
\]
where the sum is over all $n$-tuples
$\tup{j_1, j_2, \ldots, j_n} \in \calZ^n$ having the properties
that:
\begin{enumerate}
\item[\textbf{(i)}] $j_1 \preccurlyeq j_2 \preccurlyeq \cdots \preccurlyeq j_n$;
\item[\textbf{(ii)}] if $j_k = j_{k+1} = +s$ for some $s \in \calN$,
then $\pi_k < \pi_{k+1}$;
\item[\textbf{(iii)}] if $j_k = j_{k+1} = -s$ for some $s \in \calN$,
then $\pi_k > \pi_{k+1}$.
\end{enumerate}

\item This $\Gamma_{\calZ} \tup{\pi}$ will serve as an
analogue of $F_{\Comp\pi}$.
}

\pause

\item \textbf{Proposition.}
Let $w$ be a finite totally ordered set with ground
set $W$. Let $n=\left\vert W\right\vert $. Let $\overline{w}$ be the unique
poset isomorphism $w\rightarrow\left[  n\right]  $. Let $\gamma:W\rightarrow
\left\{  1,2,3,\ldots\right\}  $ be any injective map. Then, $\Gamma
_{\mathcal{Z}}\left(  w,\gamma\right)  =\Gamma_{\mathcal{Z}}\left(
\gamma\circ\overline{w}^{-1}\right)  $.

\item Again, this follows the roadmap of classical $P$-partition theory.

\pause

\item \textbf{Corollary.}
Let $\left(  P,\gamma\right)  $ be a labeled poset. Let
$n=\left\vert P\right\vert $. Then,
\[
\Gamma_{\mathcal{Z}}\left(  P,\gamma\right)  =\sum_{\substack{x:P\rightarrow
\left[  n\right]  \\\text{bijective poset}\\\text{homomorphism}}%
}\Gamma_{\mathcal{Z}}\left(  \gamma\circ x^{-1}\right)  .
\]

\pause

\item Thus, the $\Gamma_{\calZ}$ of any labeled poset can be
described in terms of the $\Gamma_{\calZ} \tup{\pi}$.

\end{itemize}

\vspace{5cm}

\end{frame}

\begin{frame}
\fti{The product formula for the $\Gamma_{\calZ}\tup{P,\gamma}$}

\begin{itemize}

\item Combining the above results, we see: \\
      \textbf{Theorem.} Let $\pi$ and $\sigma$ be two disjoint
      permutations. Then,
\[
\Gamma_{\mathcal{Z}}\left(  \pi\right)  \cdot 
\Gamma_{\mathcal{Z}} \left(
\sigma\right)  =\sum_{\tau\in S\left(  \pi,\sigma\right)  }\Gamma
_{\mathcal{Z}}\left(  \tau\right)  .
\]      

\pause

\item This generalizes the
      \[
      F_{\Comp \pi} \cdot F_{\Comp \sigma}
      = \sum\limits_{\tau \in S\tup{\pi, \sigma}} F_{\Comp \tau}
      \]
      formula in $\QSym$
      (which you can recover by setting $\calN = \NN$
      and
      $\calZ = \NN \times \set{+} = \set{+0 \prec +1 \prec +2 \prec \cdots}$).

\item Likewise, you can recover similar results by Stembridge
      and Petersen from this.

\end{itemize}

\end{frame}

\begin{frame}
\fti{Customizing the setting for $\Epk$}

\begin{itemize}

\item Remember: we want to show $\Epk$ is shuffle-compatible.

\item Specialize the above setting as follows:
\begin{itemize}
\item
      Set $\calN = \set{0, 1, 2, \ldots} \cup \set{\infty}$,
      with total order given by $0\prec1\prec2\prec\cdots \prec\infty$.
\item
      Set
      \begin{align*}
\mathcal{Z}  &  =\left(  \mathcal{N}\times\left\{  +,-\right\}  \right)
\setminus\left\{  -0,+\infty\right\} \\
&  =\left\{  +0\right\}  \cup\left\{  +n\ \mid\ n\in\left\{  1,2,3,\ldots
\right\}  \right\}  \\
& \qquad \cup\left\{  -n\ \mid\ n\in\left\{  1,2,3,\ldots\right\}
\right\}  \cup\left\{  -\infty\right\}  .
\end{align*}
Recall that the total order on $\mathcal{Z}$ has
\[
+0\prec-1\prec+1\prec-2\prec+2\prec\cdots\prec-\infty.
\]
\end{itemize}

\end{itemize}

\end{frame}

\begin{frame}
\fti{Fiber-ends}

\begin{itemize}

\item Let $n\in\mathbb{N}$. Let $g:\left[  n\right]
\rightarrow\mathcal{N}$ be any map. We define a subset
\defn{$\operatorname*{FE}\left(  g\right)  $}
of $\left[  n\right]  $ by
\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\}  \right\}
\\
&  \ \ \ \ \ \ \ \ \ \ \cup\left\{  \max\left(  g^{-1}\left(  h\right)
\right)  \ \mid\ h\in\left\{  0,1,2,3,\ldots\right\}  \right\}
\end{align*}
(ignore the maxima/minima of empty fibers). \\
In other words, $\operatorname*{FE}\left(  g\right)  $ is the set comprising
\begin{itemize}
\item
the smallest elements of all nonempty fibers of $g$ except for $g^{-1}\left(
0\right)  $ as well as
\item the largest elements of all nonempty fibers of $g$
except for $g^{-1}\left(  \infty\right)  $.
\end{itemize}
% We shall refer to the elements of
% $\operatorname*{FE}\left(  g\right)  $ as the \defn{fiber-ends} of $g$.

\end{itemize}

\end{frame}



\begin{frame}
\fti{$K$-series}

\begin{itemize}

\item Let $n\in\mathbb{N}$. If $\Lambda$ is any subset of $\left[
n\right]  $, then we define a power series \defn{$K_{n,\Lambda}^{\mathcal{Z}}%
\in\operatorname*{Pow}\mathcal{N}$} by%
\[
K_{n,\Lambda}^{\mathcal{Z}}=\sum_{\substack{g:\left[  n\right]  \rightarrow
\mathcal{N}\text{ is}\\\text{weakly increasing;}\\\Lambda\subseteq
\operatorname*{FE}\left(  g\right)  }}2^{\left\vert g\left(  \left[  n\right]
\right)  \cap\left\{  1,2,3,\ldots\right\}  \right\vert }
x_{g\tup{1}} x_{g\tup{2}} \cdots x_{g\tup{n}} .
\]

\item \textbf{Proposition.} Let $n \in \NN$. Let $\pi$ be an $n$-permutation.
Then,
\[
\Gamma_{\mathcal{Z}}\left(  \pi\right)  =K_{n,\Epk \pi}^{\mathcal{Z}}.
\]

\only<1>{
This is proven by a counting argument (if a map $g$
comes from an $\tup{\ive{n},\pi}$-partition, then
the fibers of $g$ subdivide $\ive{n}$ into intervals
on which $\pi$ is ``V-shaped''; a peak can only occur
at a border between two such intervals).
}

\pause

\item Thus, the product formula above specializes to
\[
K_{n,\operatorname*{Epk}\pi}^{\mathcal{Z}}\cdot K_{m,\operatorname*{Epk}%
\sigma}^{\mathcal{Z}}=\sum_{\tau\in S\left(  \pi,\sigma\right)  }%
K_{n+m,\operatorname*{Epk}\tau}^{\mathcal{Z}}.
\]

\item To prove that $\Epk$ is shuffle-compatible,
      we need this formula, but
      we \textbf{also}
      need to show that the ``relevant'' $K_{n,\Lambda}^{\mathcal{Z}}$
      are linearly independent.

\pause

\item Not all $K_{n,\Lambda}^{\mathcal{Z}}$ are linearly independent.
      Rather, we need to pick the right subset.

\end{itemize}

\end{frame}


\begin{frame}
\fti{Lacunar subsets and linear independence}

\begin{itemize}

\item A set $S$ of integers is called \defn{lacunar} if it contains
      no two consecutive integers.

\item \textbf{Well-known fact:} The number of lacunar subsets of
      $\ive{n}$ is the Fibonacci number $f_{n+1}$.

\pause

\item \textbf{Lemma.}
      For each permutation $\pi$, the set $\Epk \pi$ is a nonempty
      lacunar subset of $\ive{n}$. \\
      (And conversely -- although we won't need it --,
      any such subset has the form $\Epk \pi$ for some $\pi$.)

\pause

\item \textbf{Lemma.}
      The family%
\[
\left(  K_{n,\Lambda}^{\mathcal{Z}}\right)  _{n\in\mathbb{N};\ \Lambda
\subseteq\left[  n\right]  \text{ is lacunar and nonempty}}%
\]
is $\mathbb{Q}$-linearly independent.

\item This actually takes work to prove. But once proven, it completes
      the argument for the shuffle-compatibility of $\Epk$.

\end{itemize}

\end{frame}


\begin{frame}
\fti{The kernel $\calK_{\Epk}$}

\begin{itemize}

\item Recall: The \defn{kernel $\calK_{\st}$} of a descent statistic $\st$
      is the $\QQ$-vector subspace of $\QSym$ spanned by all
      differences of the form $F_\alpha - F_\beta$, with $\alpha$
      and $\beta$ being two $\st$-equivalent compositions:
      \[
      \calK_{\st} = \left< F_\alpha - F_\beta \ 
                              \mid \ \abs{\alpha} = \abs{\beta} \text{ and }
                                   \st \alpha = \st \beta \right>_\QQ .
      \]

\pause

\item Since $\Epk$ is shuffle-compatible, its kernel $\calK_{\Epk}$
      is an ideal of $\QSym$.
      How can we describe it?

\item Two ways: using the $F$-basis and using the $M$-basis.

\end{itemize}

\end{frame}


\begin{frame}
\fti{The kernel $\calK_{\Epk}$ in terms of the $F$-basis}

\begin{itemize}

\item If $J=\left(  j_{1},j_{2},\ldots,j_{m}\right)  $ and $K$
are two compositions, then we write \defn{$J\rightarrow K$} if there exists an
$\ell\in\left\{  2,3,\ldots,m\right\}  $ such that $j_{\ell}>2$ and $K=\left(
j_{1},j_{2},\ldots,j_{\ell-1},1,j_{\ell}-1,j_{\ell+1},j_{\ell+2},\ldots
,j_{m}\right)  $. \\
(In other words, we write $J\rightarrow K$ if $K$ can be
obtained from $J$ by \textquotedblleft splitting\textquotedblright\ some
non-initial entry
$j_{\ell}>2$ into two consecutive entries $1$ and $j_{\ell}-1$.)

\item \textbf{Example.} Here are all instances of the $\to$ relation
on compositions of size $\leq 5$:
\begin{align*}
\left(  1,3\right)  &\rightarrow\left(  1,1,2\right)  , \qquad
\left(  1,4\right)    \rightarrow\left(  1,1,3\right)  ,\\
\left(  1,3,1\right)   &\rightarrow\left(  1,1,2,1\right)  ,\qquad
\left(  1,1,3\right)   \rightarrow\left(  1,1,1,2\right)  ,\\
\left(  2,3\right)  & \rightarrow\left(  2,1,2\right)  .
\end{align*}

\item \textbf{Proposition.}
The ideal $\mathcal{K}_{\operatorname*{Epk}}$ of $\operatorname*{QSym}$ is
spanned (as a $\mathbb{Q}$-vector space) by all differences of the form
$F_{J}-F_{K}$, where $J$ and $K$ are two compositions satisfying $J\rightarrow
K$.

\end{itemize}

\end{frame}

\begin{frame}
\fti{The kernel $\calK_{\Epk}$ in terms of the $M$-basis}

\begin{itemize}

\item If $J=\left(  j_{1},j_{2},\ldots,j_{m}\right)  $ and $K$
are two compositions, then we write \defn{$J\underset{M}{\rightarrow}K$} if
there exists an $\ell\in\left\{  2,3,\ldots,m\right\}  $ such that $j_{\ell
}>2$ and $K=\left(  j_{1},j_{2},\ldots,j_{\ell-1},2,j_{\ell}-2,j_{\ell
+1},j_{\ell+2},\ldots,j_{m}\right)  $. (In other words, we write
$J\underset{M}{\rightarrow}K$ if $K$ can be obtained from $J$ by
\textquotedblleft splitting\textquotedblright\ some non-initial entry
$j_{\ell}>2$ into
two consecutive entries $2$ and $j_{\ell}-2$.)

\item \textbf{Example.} Here are all instances of the $\underset{M}{\rightarrow}$ relation
on compositions of size $\leq 5$:
\begin{align*}
\left(  1,3\right)  &\underset{M}{\rightarrow}\left(  1,2,1\right)  ,\qquad
\left(  1,4\right)    \underset{M}{\rightarrow}\left(  1,2,2\right)  ,\\
\left(  1,3,1\right)   &  \underset{M}{\rightarrow}\left(  1,2,1,1\right)  ,\qquad
\left(  1,1,3\right)    \underset{M}{\rightarrow}\left(  1,1,2,1\right)  ,\\
\left(  2,3\right)   &  \underset{M}{\rightarrow}\left(  2,2,1\right)  .
\end{align*}

\item \textbf{Proposition.}
The ideal $\mathcal{K}_{\operatorname*{Epk}}$ of $\operatorname*{QSym}$ is
spanned (as a $\mathbb{Q}$-vector space) by all sums of the form $M_{J}+M_{K}%
$, where $J$ and $K$ are two compositions satisfying
$J\underset{M}{\rightarrow}K$.

\end{itemize}

\end{frame}

\begin{frame}
\fti{What about other statistics?}

\begin{itemize}

\item \textbf{Question.}
      Do other descent statistics allow for similar descriptions of
      $\calK_{\st}$ ?

\pause

\item \textbf{Example.}
      \begin{align*}
      \calK_{\des}
      &= \left< F_I - F_J \ \mid \  \abs{I} = \abs{J} \text{ and } \ell\tup{I} = \ell\tup{J} \right>_{\QQ} \\
      &= \left< M_I - M_J \ \mid \  \abs{I} = \abs{J} \text{ and } \ell\tup{I} = \ell\tup{J} \right>_{\QQ}
      \end{align*}
      (where $\ell\tup{\alpha}$ denotes the length of a composition $\alpha$).

\end{itemize}

\end{frame}

\begin{frame}
\fti{Section 4}
\begin{center}
{\LARGE \bf Section 4} \\
\noindent\rule[0.5ex]{\linewidth}{1pt}
{\Large \bf Left-/right-shuffle-compatibility}
\end{center}
\vspace{1cm}
References:
\begin{itemize}
\item \href{https://github.com/darijgr/gzshuf}{\red Darij Grinberg, \textit{Shuffle-compatible permutation statistics II: the exterior peak set}, draft}.
\item \href{http://www.cip.ifi.lmu.de/~grinberg/algebra/dimcreation.pdf}{\red Darij Grinberg, \textit{Dual immaculate creation operators and a dendriform algebra structure on the quasisymmetric functions}, Canad. J. Math. 69 (2017), pp. 21--53.}
\end{itemize}
\end{frame}

\begin{frame}
\fti{Left/right-shuffle-compatibility (repeated)}

\begin{itemize}

\item We further begin the study of a finer version of
      shuffle-compatibility: ``left/right-shuffle-compatibility''.

\item Given two disjoint nonempty permutations $\pi$ and $\sigma$,
      \begin{itemize}
      \item
      a \defn{left shuffle} of $\pi$ and $\sigma$ is a shuffle
      of $\pi$ and $\sigma$ that starts with a letter of $\pi$;
      \item
      a \defn{right shuffle} of $\pi$ and $\sigma$ is a shuffle
      of $\pi$ and $\sigma$ that starts with a letter of $\sigma$.
      \end{itemize}

\item We let $S_\prec \tup{\pi, \sigma}$ be the set of all left shuffles of $\pi$
      and $\sigma$. \\
      We let $S_\succ \tup{\pi, \sigma}$ be the set of all right shuffles of $\pi$
      and $\sigma$.

\item A statistic $\st$ is said to be \defn{left-shuffle-compatible}
      if for any two disjoint nonempty permutations $\pi$ and $\sigma$
      such that
      \[
      \text{the first entry of }\pi\text{ is greater than the first entry of }%
      \sigma,
      \]
      the
      multiset
      \[
      \set{ \st\tau \mid \tau\in S_\prec \tup{\pi, \sigma} }_{\text{multiset}}
      \]
      depends only on $\st \pi$, $\st \sigma$, $\abs{\pi}$ and
      $\abs{\sigma}$. \\
      %Similarly define \defn{right-shuffle-compatible}.

\item We show that $\Des$, $\des$, $\Lpk$ and $\Epk$ are
      left- and right-shuffle-compatible.
      (But not $\maj$ or $\Rpk$.)

\end{itemize}

\end{frame}

\begin{frame}
\fti{Dendriform structure on $\QSym$, introduction}

\begin{itemize}

\item This proof will use a \textbf{dendriform algebra} structure on $\QSym$,
      as well as two other operations and a bit of the Hopf algebra
      structure. \\
      I don't know of a combinatorial proof.

\item This structure first appeared in: \\
      {\red \href{http://www.cip.ifi.lmu.de/~grinberg/algebra/dimcreation.pdf}{\red Darij Grinberg, \textit{Dual immaculate creation operators and a dendriform algebra structure on the quasisymmetric functions}, Canad. J. Math. 69 (2017), pp. 21--53.}}
      \\
      But the ideas go back to:
      \begin{itemize}
      \item
      {\red \href{http://www.sciencedirect.com/science/article/pii/0001870877900421}{Gl\^anffrwd P. Thomas, \textit{Frames, Young tableaux, and Baxter sequences}, Advances in Mathematics, Volume 26, Issue 3, December 1977, Pages 275--289}}.
      \item
      {\red \href{http://arxiv.org/abs/math/0510218}{Jean-Christophe Novelli, Jean-Yves Thibon,
\textit{Construction of dendriform trialgebras}, arXiv:math/0510218}}.
      \end{itemize}
      Something similar also appeared in:
      {\red \href{https://doi.org/10.1016/j.jpaa.2009.06.001}{Aristophanes Dimakis, Folkert M\"uller-Hoissen,
      \textit{Quasi-symmetric functions and the KP hierarchy},
      Journal of Pure and Applied Algebra, Volume 214, Issue 4, April 2010, Pages 449--460}}.

\end{itemize}

\vspace{9cm}

\end{frame}

\begin{frame}
\fti{Dendriform structure on $\QSym$, part 1}

\begin{itemize}

\item For any monomial $\mathfrak{m}$, let $\Supp \mathfrak{m}$
denote the set $\left\{i \mid x_i \text{ appears in } \mathfrak{m}\right\}$.

\item \textbf{Example.} $\Supp \left(x_3^5 x_6 x_8\right) = \left\{3,6,8\right\}$.

\only<2>{
\item We define a binary operation $\left.  \prec\right.$ on the
$\QQ$-vector space $\Powser$ as follows:

\begin{itemize}
\item On monomials, it should be given by
\[
\mathfrak{m}\left.  \prec\right.  \mathfrak{n}=\left\{
\begin{array}
[c]{c}%
\mathfrak{m}\cdot\mathfrak{n},\ \ \ \ \ \ \ \ \ \ \text{if }\min\left(
\operatorname*{Supp}\mathfrak{m}\right)  <\min\left(  \operatorname*{Supp}%
\mathfrak{n}\right)  ;\\
0,\ \ \ \ \ \ \ \ \ \ \text{if }\min\left(  \operatorname*{Supp}%
\mathfrak{m}\right)  \geq\min\left(  \operatorname*{Supp}\mathfrak{n}\right)
\end{array}
\right.
\]
for any two monomials $\mathfrak{m}$ and $\mathfrak{n}$.
\item It should be $\QQ$-bilinear.
\item It should be continuous (i.e., its $\QQ$-bilinearity also
applies to infinite $\QQ$-linear combinations).
\end{itemize}

\item Well-definedness is pretty clear.

\item \textbf{Example.} $\left(x_2^2 x_4\right) \left.\prec\right. \left(x_3^2 x_5\right) = x_2^2 x_3^2 x_4 x_5$, but
$\left(x_2^2 x_4\right) \left.\prec\right. \left(x_2^2 x_5\right) = 0$.
}
\only<3>{
\item We define a binary operation $\left.  \succeq\right.$ on the
$\QQ$-vector space $\Powser$ as follows:

\begin{itemize}
\item On monomials, it should be given by
\[
\mathfrak{m}\left.  \succeq\right.  \mathfrak{n}=\left\{
\begin{array}
[c]{c}%
\mathfrak{m}\cdot\mathfrak{n},\ \ \ \ \ \ \ \ \ \ \text{if }\min\left(
\operatorname*{Supp}\mathfrak{m}\right) \geq \min\left(  \operatorname*{Supp}%
\mathfrak{n}\right)  ;\\
0,\ \ \ \ \ \ \ \ \ \ \text{if }\min\left(  \operatorname*{Supp}%
\mathfrak{m}\right) < \min\left(  \operatorname*{Supp}\mathfrak{n}\right)
\end{array}
\right.
\]
for any two monomials $\mathfrak{m}$ and $\mathfrak{n}$.
\item It should be $\QQ$-bilinear.
\item It should be continuous (i.e., its $\QQ$-bilinearity also
applies to infinite $\QQ$-linear combinations).
\end{itemize}

\item Well-definedness is pretty clear.

\item \textbf{Example.} $\left(x_2^2 x_4\right) \left.\succeq\right. \left(x_3^2 x_5\right) = 0$, but
$\left(x_2^2 x_4\right) \left.\succeq\right. \left(x_2^2 x_5\right) = x_2^4 x_4 x_5$.
}


\end{itemize}

\vspace{9cm}

\end{frame}


\begin{frame}
\fti{Dendriform structure on $\QSym$, part 2}

\begin{itemize}

\item We now have defined two binary operations $\left.\prec\right.$
and $\left.\succeq\right.$ on $\Powser$. They satisfy:
\begin{align*}
a\left.  \prec\right.  b+a \left.  \succeq\right.b  &  =ab;\\
\left(  a\left.  \prec\right.  b\right)  \left.  \prec\right.  c  &  =a\left.
\prec\right.  \left(  bc\right)  ;\\
\left(  a \left.  \succeq\right.  b\right)  \left.  \prec\right.  c
&  =a \left.  \succeq\right. \left(  b\left.  \prec\right.
c\right)  ;\\
a \left.  \succeq\right.  \left(  b \left.  \succeq\right.
c\right)   &  =\left(  ab\right)  \left.  \succeq\right.  c.
\end{align*}

\pause

\item This says that $\left(\Powser, \left.\prec\right., \left.\succeq\right.\right)$
is a \textit{dendriform algebra} in the sense of Loday
(see, e.g., {\red \href{http://arxiv.org/abs/1101.0267}{Zinbiel,
\textit{Encyclopedia of types of algebras 2010},
arXiv:1101.0267}}).

\pause

\item $\QSym$ is closed under both operations $\prec$ and $\succeq$.
Thus, $\QSym$ becomes a dendriform subalgebra of $\Powser$.

\end{itemize}

\vspace{9cm}

\end{frame}

\begin{frame}
\fti{The kernel criterion for left/right-shuffle-compatibility}

\begin{itemize}

\only<1>{
\item Recall the \textbf{Theorem:} The descent statistic $\st$ is
      shuffle-compatible if and only if $\calK_{\st}$ is an
      ideal of $\QSym$.
}
\pause
\only<2-5>{
\item Similarly, we have:
      \begin{itemize}
      \item \textbf{Theorem.} The descent statistic $\st$ is
      left-shuffle-compatible if and only if $\calK_{\st}$ is a
      $\prec$-ideal of $\QSym$ (that is: $\QSym \prec \calK_{\st}
      \subseteq \calK_{\st}$ and $\calK_{\st} \prec \QSym
      \subseteq \calK_{\st}$).
      \item \textbf{Theorem.} The descent statistic $\st$ is
      right-shuffle-compatible if and only if $\calK_{\st}$ is a
      $\succeq$-ideal of $\QSym$ (that is: $\QSym \succeq \calK_{\st}
      \subseteq \calK_{\st}$ and $\calK_{\st} \succeq \QSym
      \subseteq \calK_{\st}$).
      \end{itemize}

\pause

\item \textbf{Corollary.} Let $\st$ be a descent statistic.
      If $\st$ has $2$ of the $3$ properties
      ``shuffle-compatible'', ``left-shuffle-compatible'' and
      ``right-shuffle-compatible'', then it has all $3$.
      \\ (To prove this, recall $ab = a \prec b + a \succeq b$.)
}

\pause

\item \only<4>{
      \textbf{Question.} Are there non-shuffle-compatible but
      left-shuffle-compatible descent statistics?
      \\ (I don't know of any, but haven't looked far.)
      }
      \pause
      \only<5>{
      Okay, but how do we actually prove that $\calK_{\st}$
      is a $\prec$-ideal of $\QSym$ ?
      }
\end{itemize}

\vspace{10cm}

\end{frame}

\begin{frame}
\fti{The dendriform product formula for the $F_\alpha$}

\begin{itemize}

\item An analogue of the product formula for $F_{\Comp\pi} \cdot F_{\Comp\sigma}$:
      \textbf{Theorem.} Let $\pi$ and $\sigma$ be two disjoint nonempty
        permutations. Assume that%
        \[
        \text{the first entry of }\pi\text{ is greater than the first entry of }%
        \sigma\text{.}%
        \]
        Then,%
        \[
        F_{\operatorname*{Comp}\pi}\left.  \prec\right.  F_{\operatorname*{Comp}%
        \sigma}=\sum_{\tau\in S_{\prec}\left(  \pi,\sigma\right)  }%
        F_{\operatorname*{Comp}\tau}%
        \]
        and%
        \[
        F_{\operatorname*{Comp}\pi}\left.  \succeq\right.  F_{\operatorname*{Comp}%
        \sigma}=\sum_{\tau\in S_{\succ}\left(  \pi,\sigma\right)  }%
        F_{\operatorname*{Comp}\tau}.
        \]

\pause

\item This theorem yields that $\Des$ is left-shuffle-compatible
      and right-shuffle-compatible,
      just as the product formula showed that $\Des$ is
      shuffle-compatible.

\pause

\item Can we play the same game with $\Epk$, using our
      $K_{n,\Lambda}^{\mathcal{Z}}$ series instead of $F_\alpha$ ?
      \pause
      \\ I don't know how. Instead, I use a different approach.

\end{itemize}
\vspace{10cm}
\end{frame}

\begin{frame}
\fti{The $\bel$ and $\tvi$ operations}

\begin{itemize}

\only<1-2>{
\item I need two other operations on quasisymmetric functions.
}

\only<1>{
\item We define a binary operation $\bel$ on the
$\QQ$-vector space $\Powser$ as follows:

\begin{itemize}
\item On monomials, it should be given by
\[
\mathfrak{m}\bel \mathfrak{n}=\left\{
\begin{array}
[c]{c}%
\mathfrak{m}\cdot\mathfrak{n},\ \ \ \ \ \ \ \ \ \ \text{if }\max\left(
\operatorname*{Supp}\mathfrak{m}\right)  \leq \min\left(  \operatorname*{Supp}%
\mathfrak{n}\right)  ;\\
0,\ \ \ \ \ \ \ \ \ \ \text{if }\max\left(  \operatorname*{Supp}%
\mathfrak{m}\right)  >\min\left(  \operatorname*{Supp}\mathfrak{n}\right)
\end{array}
\right.
\]
for any two monomials $\mathfrak{m}$ and $\mathfrak{n}$.
\item It should be $\QQ$-bilinear.
\item It should be continuous (i.e., its $\QQ$-bilinearity also
applies to infinite $\QQ$-linear combinations).
\end{itemize}

\item Well-definedness is pretty clear.

\item \textbf{Example.} $\left(x_2^2 x_4\right) \bel \left(x_4^2 x_5\right) = x_2^2 x_4^3 x_5$ and
$\left(x_2^2 x_4\right) \bel \left(x_3^2 x_5\right) = 0$.
}
\only<2>{
\item We define a binary operation $\tvi$ on the
$\QQ$-vector space $\Powser$ as follows:

\begin{itemize}
\item On monomials, it should be given by
\[
\mathfrak{m}\tvi  \mathfrak{n}=\left\{
\begin{array}
[c]{c}%
\mathfrak{m}\cdot\mathfrak{n},\ \ \ \ \ \ \ \ \ \ \text{if }\max\left(
\operatorname*{Supp}\mathfrak{m}\right) < \min\left(  \operatorname*{Supp}%
\mathfrak{n}\right)  ;\\
0,\ \ \ \ \ \ \ \ \ \ \text{if }\max\left(  \operatorname*{Supp}%
\mathfrak{m}\right) \geq \min\left(  \operatorname*{Supp}\mathfrak{n}\right)
\end{array}
\right.
\]
for any two monomials $\mathfrak{m}$ and $\mathfrak{n}$.
\item It should be $\QQ$-bilinear.
\item It should be continuous (i.e., its $\QQ$-bilinearity also
applies to infinite $\QQ$-linear combinations).
\end{itemize}

\item Well-definedness is pretty clear.

\item \textbf{Example.} $\left(x_2^2 x_4\right) \tvi \left(x_4^2 x_5\right) = 0$, but
$\left(x_2^2 x_4\right) \tvi \left(x_5^2 x_6\right) = x_2^2 x_4 x_5^2 x_6$.
}

\only<3->{
\item $\QSym$ is closed under both operations $\bel$ and $\tvi$.

\item Belgthor ($\bel$) and Tvimadur ($\tvi$) are calendar runes
(for two of the 19 years of the Metonic cycle). \\
I sought two (unused) symbols that (roughly) look like ``stacking one
thing (monomial) atop another'', \\
allowing overlap ($\bel$) and
disallowing overlap ($\tvi$).
}

\end{itemize}

\vspace{9cm}

\end{frame}

\begin{frame}
\fti{A crucial identity}

\begin{itemize}

\item \textbf{Proposition.} For any $a \in \Powser$
and $b \in \QSym$, we have
\[
\sum_{\left(  b\right)  }\left(  S\left(  b_{\left(  1\right)  }\right)
\bel a\right)  b_{\left(  2\right)  }=a\left.  \prec\right.  b ,
\]
where we use the Hopf algebra structure on $\QSym$
\only<1>{ and the
following notations:
\begin{itemize}
\item $S$ for the antipode of $\QSym$;
\item Sweedler's notation $\sum\limits_{\left(
b\right)  }b_{\left(  1\right)  }\otimes b_{\left(  2\right)  }$ for
$\Delta\left(  b\right)  $.
\end{itemize}
}
\only<2->{.}

\only<2>{
Restatement without using Hopf algebras:
\[
\sum_{p=0}^k
\tup{-1}^p
\tup{ M'_{\alpha_p, \alpha_{p-1}, \ldots, \alpha_1} \bel a }
M_{\alpha_{p+1}, \alpha_{p+2}, \ldots, \alpha_k}
= a \left. \prec \right. M_\alpha
\]
for any composition $\alpha = \tup{\alpha_1, \alpha_2, \ldots, \alpha_k}$
and any $a \in \Powser$, where
\[
M'_{\alpha_p, \alpha_{p-1}, \ldots, \alpha_1}
= \sum_{i_1 \leq i_2 \leq \cdots \leq i_p} x_{i_1}^{\alpha_1} x_{i_2}^{\alpha_2} \cdots x_{i_p}^{\alpha_p} .
\]
}

\pause \pause

\item This proposition was important in my study of ``dual
      immaculate creation operators''; it is equally helpful
      here. \\
      \textbf{Corollary.} Let $M$ be an ideal of $\QSym$.
      If $\QSym \bel  M\subseteq M$, then $M\left.  \prec\right.  \QSym \subseteq M$.

\pause

\item A similar identity for $\tvi$ yields: \\
      \textbf{Corollary.} Let $M$ be an ideal of $\QSym$.
      If $\QSym \tvi  M\subseteq M$, then $\QSym \left.  \succeq\right.  M \subseteq M$.

\pause

\item \textbf{Corollary.} Let $M$ be an ideal of $\QSym$
      that is a left $\bel$-ideal (that is,
      $\QSym \bel  M\subseteq M$) and a left $\tvi$-ideal
      (that is, $\QSym \tvi  M\subseteq M$).
      Then, $M$ is a $\prec$-ideal and a $\succeq$-ideal
      of $\QSym$.
\end{itemize}

\vspace{9cm}

\end{frame}


\begin{frame}
\fti{``Runic calculus''}

\begin{itemize}

\item The operations $\bel$ and $\tvi$ are associative and unital (with unity $1$).
\pause

\only<2>{
\item For any two nonempty (i.e., $\neq \tup{}$) compositions $\alpha$ and $\beta$, we have
\begin{align*}
M_{\alpha}\bel  M_{\beta} &=M_{\left[  \alpha,\beta\right]  }+M_{\alpha
\odot\beta}; \\
M_{\alpha} \tvi  M_{\beta} &=M_{\left[  \alpha,\beta\right] } ; \\
F_{\alpha} \bel  F_{\beta} &=F_{\alpha\odot\beta} ; \\
F_{\alpha} \tvi  F_{\beta} &= F_{\left[  \alpha,\beta\right]  } ,
\end{align*}
where $\left[  \alpha,\beta\right]  $ and $\alpha\odot\beta$ are
two compositions defined by%
\begin{align*}
& \left[  \left(  \alpha_{1},\alpha_{2},\ldots,\alpha_{\ell}\right)  ,\left(
\beta_{1},\beta_{2},\ldots,\beta_{m}\right)  \right]  \\
 &  =\left(  \alpha
_{1},\alpha_{2},\ldots,\alpha_{\ell},\beta_{1},\beta_{2},\ldots,\beta
_{m}\right)
\end{align*}
and
\begin{align*}
&\left(  \alpha_{1},\alpha_{2},\ldots,\alpha_{\ell}\right)  \odot\left(
\beta_{1},\beta_{2},\ldots,\beta_{m}\right)   \\
&  =\left(  \alpha_{1}%
,\alpha_{2},\ldots,\alpha_{\ell-1},\alpha_{\ell}+\beta_{1},\beta_{2},\beta
_{3},\ldots,\beta_{m}\right)  .
\end{align*}
}

\only<3>{
\item They satisfy
\begin{align*}
\left(  a\left.
%TCIMACRO{\TeXButton{bel}{\textarm{\belgthor}}}%
%BeginExpansion
\textarm{\belgthor}%
%EndExpansion
\right.  b\right)  \left.
%TCIMACRO{\TeXButton{tvi}{\textarm{\tvimadur}}}%
%BeginExpansion
\textarm{\tvimadur}%
%EndExpansion
\right.  c-a\left.
%TCIMACRO{\TeXButton{bel}{\textarm{\belgthor}}}%
%BeginExpansion
\textarm{\belgthor}%
%EndExpansion
\right.  \left(  b\left.
%TCIMACRO{\TeXButton{tvi}{\textarm{\tvimadur}}}%
%BeginExpansion
\textarm{\tvimadur}%
%EndExpansion
\right.  c\right)   &  =\varepsilon\left(  b\right)  \left(  a\left.
%TCIMACRO{\TeXButton{tvi}{\textarm{\tvimadur}}}%
%BeginExpansion
\textarm{\tvimadur}%
%EndExpansion
\right.  c-a\left.
%TCIMACRO{\TeXButton{bel}{\textarm{\belgthor}}}%
%BeginExpansion
\textarm{\belgthor}%
%EndExpansion
\right.  c\right)  ;\\
\left(  a\left.
%TCIMACRO{\TeXButton{tvi}{\textarm{\tvimadur}}}%
%BeginExpansion
\textarm{\tvimadur}%
%EndExpansion
\right.  b\right)  \left.
%TCIMACRO{\TeXButton{bel}{\textarm{\belgthor}}}%
%BeginExpansion
\textarm{\belgthor}%
%EndExpansion
\right.  c-a\left.
%TCIMACRO{\TeXButton{tvi}{\textarm{\tvimadur}}}%
%BeginExpansion
\textarm{\tvimadur}%
%EndExpansion
\right.  \left(  b\left.
%TCIMACRO{\TeXButton{bel}{\textarm{\belgthor}}}%
%BeginExpansion
\textarm{\belgthor}%
%EndExpansion
\right.  c\right)   &  =\varepsilon\left(  b\right)  \left(  a\left.
%TCIMACRO{\TeXButton{bel}{\textarm{\belgthor}}}%
%BeginExpansion
\textarm{\belgthor}%
%EndExpansion
\right.  c-a\left.
%TCIMACRO{\TeXButton{tvi}{\textarm{\tvimadur}}}%
%BeginExpansion
\textarm{\tvimadur}%
%EndExpansion
\right.  c\right)  ,
\end{align*}
where $\varepsilon:\Powser\to\QQ$ sends $f$ to $f\left(0,0,0,\ldots\right)$.

\pause

\item As a consequence,
\[
\left(  a\left.
%TCIMACRO{\TeXButton{bel}{\textarm{\belgthor}}}%
%BeginExpansion
\textarm{\belgthor}%
%EndExpansion
\right.  b\right)  \left.
%TCIMACRO{\TeXButton{tvi}{\textarm{\tvimadur}}}%
%BeginExpansion
\textarm{\tvimadur}%
%EndExpansion
\right.  c+\left(  a\left.
%TCIMACRO{\TeXButton{tvi}{\textarm{\tvimadur}}}%
%BeginExpansion
\textarm{\tvimadur}%
%EndExpansion
\right.  b\right)  \left.
%TCIMACRO{\TeXButton{bel}{\textarm{\belgthor}}}%
%BeginExpansion
\textarm{\belgthor}%
%EndExpansion
\right.  c=a\left.
%TCIMACRO{\TeXButton{bel}{\textarm{\belgthor}}}%
%BeginExpansion
\textarm{\belgthor}%
%EndExpansion
\right.  \left(  b\left.
%TCIMACRO{\TeXButton{tvi}{\textarm{\tvimadur}}}%
%BeginExpansion
\textarm{\tvimadur}%
%EndExpansion
\right.  c\right)  +a\left.
%TCIMACRO{\TeXButton{tvi}{\textarm{\tvimadur}}}%
%BeginExpansion
\textarm{\tvimadur}%
%EndExpansion
\right.  \left(  b\left.
%TCIMACRO{\TeXButton{bel}{\textarm{\belgthor}}}%
%BeginExpansion
\textarm{\belgthor}%
%EndExpansion
\right.  c\right)  .
\]
This says that $\left(\QSym, \bel, \tvi\right)$ is a
$As^{\left\langle 2\right\rangle }$\textit{-algebra}
(in the sense of Loday).

\item \textbf{Question.} What other identities do
$\bel$, $\tvi$, $\prec$ and $\succeq$ satisfy?
}

\end{itemize}

\vspace{9cm}

\end{frame}


\begin{frame}
\fti{How to check left-/right-shuffle-compatibility}

\begin{itemize}

\item Recall the \textbf{Corollary:} Let $M$ be an ideal of $\QSym$
      that is a left $\bel$-ideal (that is,
      $\QSym \bel  M\subseteq M$) and a left $\tvi$-ideal
      (that is, $\QSym \tvi  M\subseteq M$).
      Then, $M$ is a $\prec$-ideal and a $\succeq$-ideal
      of $\QSym$.

\pause

\item Given a shuffle-compatible
      descent statistic $\st$, we thus conclude
      that if $\calK_{\st}$ is a left $\bel$-ideal and a left
      $\tvi$-ideal, then $\st$ is left-shuffle-compatible
      and right-shuffle-compatible.

\pause

\item Fortunately, this is easy to apply: \\
      \textbf{Proposition.}
      Let $\operatorname*{st}$ be a descent statistic.
      
      \begin{itemize}
      \item $\calK_{\st}$ is a left $\bel$-ideal
of $\operatorname*{QSym}$ if and only if
\only<3>{
$\operatorname*{st}$ has the
following property: If $J$ and $K$ are two $\operatorname*{st}$-equivalent
nonempty compositions, and if $G$ is any nonempty composition, then $G\odot J$
and $G\odot K$ are $\operatorname*{st}$-equivalent.
}
\only<4>{for each fixed nonempty composition $A$,
the value
$\st\tup{A \odot B}$ (for a nonempty composition $B$)
is uniquely determined by $\abs{B}$ and $\st B$.
}
      \item $\calK_{\st}$ is a left $\tvi$-ideal
of $\operatorname*{QSym}$ if and only if
\only<3>{
$\operatorname*{st}$ has the
following property: If $J$ and $K$ are two $\operatorname*{st}$-equivalent
nonempty compositions, and if $G$ is any nonempty composition, then $\left[
G,J\right]  $ and $\left[  G,K\right]  $ are $\operatorname*{st}$-equivalent.
}
\only<4>{for each fixed nonempty composition $A$,
the value
$\st\tup{\ive{A, B}}$ (for a nonempty composition $B$)
is uniquely determined by $\abs{B}$ and $\st B$.
}
      \end{itemize}

\end{itemize}

\vspace{5cm}

\end{frame}

\begin{frame}
\fti{Why $\Epk$ is left- and right-shuffle-compatible}

\begin{itemize}

\item Thus, proving that $\Epk$ is left- and right-shuffle-compatible
      requires showing that $\Epk\tup{A \odot B}$ and $\Epk\tup{\ive{A, B}}$
      (for nonempty compositions $A$ and $B$) are uniquely determined
      by $\abs{B}$ and $\Epk B$ when $A$ is fixed.

\pause

\item This is not hard:
\begin{align*}
\operatorname*{Epk}\left(  A\odot B\right)  &=\left(
\left(  \operatorname*{Epk}A\right)  \setminus\left\{  n\right\}  \right)
\cup\left(  \operatorname*{Epk}B+n\right)  ; \\
\operatorname*{Epk}\left(  \left[  A,B\right]  \right)
&=\left(  \operatorname*{Epk}A\right)  \cup\left(  \left(  \operatorname*{Epk}%
B+n\right)  \setminus\left\{  n+1\right\}  \right) ,
\end{align*}
where $n = \abs{A}$.

\pause

\item Similarly,
      \only<3>{
        \begin{itemize}
        \item $\Des$ is left- and right-shuffle-compatible (again);
        \item $\des$ is left- and right-shuffle-compatible;
        \item $\maj$ is \textbf{not} left- or right-shuffle-compatible
              ($\maj\tup{A \odot B}$ and $\maj\tup{\ive{A, B}}$ depend
              not just on $\abs{A}$, $\abs{B}$, $\maj A$ and $\maj B$,
              but also on $\des B$).
        \end{itemize}
      }
      \pause
      \only<4>{
        \begin{itemize}
        \item $\tup{\des, \maj}$ is left- and right-shuffle-compatible;
        \item $\Lpk$ is left- and right-shuffle-compatible;
        \item $\Rpk$ is \textbf{not} left- or right-shuffle-compatible;
        \item $\Pk$ is \textbf{not} left- or right-shuffle-compatible.
        \end{itemize}
      }

\item More statistics remain to be analyzed.

\end{itemize}

\vspace{10cm}

\end{frame}

\begin{frame}
\fti{Further questions}

\begin{itemize}

\item \textbf{Question (repeated).}
      Can a statistic be shuffle-compatible without being
      a descent statistic? \\
      (Would $\operatorname{FQSym}$ help in studying such
      statistics?)

\item \textbf{Question (repeated).}
      Can a descent statistic be left-shuffle-compatible without
      being shuffle-compatible?

\item \textbf{Question.}
      What mileage do we get out of $\calZ$-enriched
      $\tup{P,\gamma}$-partitions for other choices of
      $\calN$ and $\calZ$ ?

\item \textbf{Question (repeated).}
      Where do the $\Gamma_{\calZ}\tup{P,\gamma}$ live?

\item \textbf{Question.}
      Hsiao and Petersen have generalized enriched
      $\tup{P, \gamma}$-partitions to ``colored
      $\tup{P, \gamma}$-partitions'' (with $\set{+,-}$
      replaced by an $m$-element set).
      Does this generalize our results?

\end{itemize}

\end{frame}



% \begin{frame}
% \fti{}

% \begin{itemize}

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% \end{itemize}

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\begin{frame}
\fti{Thanks}

\textbf{Thanks} to Ira Gessel and Yan Zhuang for initiating this direction
(and for helpful discussions), and to
Alex Yong for an invitation to UIUC. \\
And thanks to you for attending!

\vspace{3cm}

\textbf{slides: \red \url{http://www.cip.ifi.lmu.de/~grinberg/algebra/urbana18b.pdf}} \\
\textbf{paper: \red \url{http://www.cip.ifi.lmu.de/~grinberg/algebra/gzshuf2.pdf}} \\
\textbf{project: \red \url{https://github.com/darijgr/gzshuf}}

\end{frame}

\end{document}
