A list of errata in the paper "Counting permutations with given cycle structure and descent set" by Ira M. Gessel and Christophe Reutenauer ( https://doi.org/10.1016/0097-3165(93)90095-P ):
- page 191: "we already considered" -> "were already considered".
- page 194: Before "We denote by QSym the algebra of quasi-symmetric
functions", add "A quasi-symmetric power series f will be called a
quasi-symmetric function if it is of bounded degree (i.e., there
exists an integer d such that all monomials that appear in f have
degree < d).".
The "bounded degree" condition is important, because without it,
QSym would contain power series like the infinite product
(1 + x_1) (1 + x_2) (1 + x_3) ..., and then {F_C} would not be a
basis of QSym.
- page 194: In the sentence that begins with "Define a function",
the "|->" arrow should be an "->" arrow (i.e., it should not have
a serif on its left end).
- page 194: "lexocigraphic" -> "lexicographic" (ignore unless you have
a preprint around).
- page 196: The mapping $U$ is not obviously well-defined. Indeed, the
word "necklace" means "primitive necklace" in this paper, and so the
definition of $U(w)$ requires checking that the $v_c(w)$ actually are
(primitive) necklaces. This is true but not obvious.
Here is a sneaky way to tweak the argument so that this gap
disappears. Namely, make $U$ a *partial* map; the definition remains
the same, except that in the case when not all of the $v_c(w)$ are
necklaces, we simply leave $U(w)$ undefined. The proof of Lemma 3.4
still shows that every finite multiset $M$ of primitive necklaces of
length $n$ can be written in the form $U(w)$ for some word $w$. Thus,
if the map $U$ is not total, then there must be strictly more words of
a given evaluation than there are finite multisets of primitive
necklaces of this evaluation. But this would be absurd, since the
Chen-Fox-Lyndon factorization ensures that the number of words of a
given evaluation equals the number of multisets of primitive necklaces
of this evaluation. Hence, the map $U$ is total, and everything works
as intended.
- page 202: The continuous homomorphism Lambda_m is not actually
well-defined on all formal power series in x_1, x_2, ...; for example,
if m > 0, then it would have to send the formal power series
1 + x_1 + x_1^2 + x_1^3 + ... to 1 + 1 + 1 + 1 + ..., which is absurd.
Fortunately, it is well-defined on all formal power series of bounded
degree in x_1, x_2, ....
- page 213, Theorem 9.5: Not a mistake, just a remark: This is easy to
prove without quasisymmetric functions, too. I gave it as a homework
exercise in my combinatorics class (see Proposition 0.13 in
http://www.cip.ifi.lmu.de/~grinberg/t/17f/hw6os.pdf ).