11/13/2025
By Joris Roos
Title: Descents of permutations with only even or only odd cycles
Date: Wednesday, December 3
Time: 11 a.m. to noon
Room: Southwick 350W
Everyone is welcome!
Abstract: It is known that, when n is even, the number of permutations of {1,2,...,n} all of whose cycles have odd length equals the number of those all of whose cycles have even length. Adin, Hegedüs and Roichman recently found a surprising refinement of this equality, showing that it still holds when restricting to permutations with a given descent set J on one side, and permutations with ascent set J on the other. Their proof is algebraic and uses heavy machinery. It also yields a version for odd n.
In this talk we give a bijective proof of their result. First, using some beautiful bijections of Gessel, Reutenauer and others, we restate it in terms of multisets of necklaces, which we interpret as words. Then, we construct a bijection between words all of whose Lyndon factors have odd length and are distinct, and words all of whose Lyndon factors have even length.
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