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 ).