Descent sets of cyclic permutations

Details Descent sets of cyclic permutations Descent sets of cyclic permutations

2009-06-15

arxiv.org/abs/0906.2795v2

Adv. in Appl. Math. 47 (2011), 688-709

Mathematics

Combinatorics

22 pages, final journal version

Scientific paper

We present a bijection between cyclic permutations of {1,2,...,n+1} and permutations of {1,2,...,n} that preserves the descent set of the first n entries and the set of weak excedances. This non-trivial bijection involves a Foata-like transformation on the cyclic notation of the permutation, followed by certain conjugations. We also give an alternate derivation of the consequent result about the equidistribution of descent sets using work of Gessel and Reutenauer. Finally, we prove a conjecture of the author in [SIAM J. Discrete Math. 23 (2009), 765-786] and a conjecture of Eriksen, Freij and W\"astlund.

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