Geodesic
The Number of Cycles with a Given Descent Set
Séminaire lotharingien de combinatoire, 80B (2018) 
Sergi Elizalde; Justin M. Troyka. The Number of Cycles with a Given Descent Set. Séminaire lotharingien de combinatoire, 80B (2018). http://geodesic.mathdoc.fr/item/SLC_2018_80B_a7/
@article{SLC_2018_80B_a7,
     author = {Sergi Elizalde and Justin M. Troyka},
     title = {The {Number} of {Cycles} with a {Given} {Descent} {Set}},
     journal = {S\'eminaire lotharingien de combinatoire},
     year = {2018},
     volume = {80B},
     url = {http://geodesic.mathdoc.fr/item/SLC_2018_80B_a7/}
}
TY  - JOUR
AU  - Sergi Elizalde
AU  - Justin M. Troyka
TI  - The Number of Cycles with a Given Descent Set
JO  - Séminaire lotharingien de combinatoire
PY  - 2018
VL  - 80B
UR  - http://geodesic.mathdoc.fr/item/SLC_2018_80B_a7/
ID  - SLC_2018_80B_a7
ER  - 
%0 Journal Article
%A Sergi Elizalde
%A Justin M. Troyka
%T The Number of Cycles with a Given Descent Set
%J Séminaire lotharingien de combinatoire
%D 2018
%V 80B
%U http://geodesic.mathdoc.fr/item/SLC_2018_80B_a7/
%F SLC_2018_80B_a7

Voir la notice de l'acte provenant de la source Séminaire Lotharingien de Combinatoire website

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 contained in [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.