Exact and asymptotic enumeration of cyclic permutations according to descent set

Abstract

Using a result of Gessel and Reutenauer, we find a simple formula for the number of cyclic permutations with a given descent set, by expressing it in terms of ordinary descent numbers (i.e., those counting all permutations with a given descent set). We then use this formula to show that, for almost all sets I ⊆ [n − 1], the fraction of size-n permutations with descent set I which are n-cycles is asymptotically 1/n. As a special case, we recover a result of Stanley for alternating cycles. We also use our formula to count n-cycles with no two consecutive descents.

Higher Lie characters and cyclic descent extension on conjugacy classes

“…Proof. Let H(x, q) := The reader is referred to [12] for further discussion and relations to the enumeration of Lyndon words.…”

Enumerating two permutation classes by the number of cycles

“…those avoiding 312 and 213 [18,15]; other pairs of length 3 permutations [10,6]; almost-increasing permutations, which avoid four length 4 permutations [11,2]; and permutations from certain grid classes [16,8,1,3]. In addition the numbers of cyclic permutations of length n avoiding 123 or 321 as a consecutive pattern are given in [14]. There are also many results regarding pattern-avoiding involutions and fixed points of pattern-avoiding permutations [17,13,7,5,12].…”

Two first-order logics of permutations

“…In the huge combinatorics literature on permutations, there are, to our knowledge, very few works that consider at the same time the algebraic and the pattern view point. Some of those we are aware of study the stability of permutation classes by composition of permutations, as in [16], while others are interested in cycles exhibiting a particular pattern structure, see for example [5,10] and references therein.…”