Geodesic
The Number of Cycles with a Given Descent Set
Séminaire lotharingien de combinatoire, 80B (2018)

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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. 

@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},
     publisher = {mathdoc},
     volume = {80B},
     year = {2018},
     url = {http://geodesic.mathdoc.fr/item/SLC_2018_80B_a7/}
}
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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/