Ballot permutations and odd order permutations

Abstract

A permutation π is ballot if, for all k, the word π1⋯πk has at least as many ascents as it has descents. Let b(n) denote the number of ballot permutations of size n, and let p(n) denote the number of permutations which have odd order in the symmetric group Sn. Callan conjectured that b(n)=p(n) for all n, which was proved by Bernardi, Duplantier, and Nadeau.We propose a refinement of Callan’s original conjecture. Let b(n,d) denote the number of ballot permutations with d descents. Let p(n,d) denote the number of odd order permutations with M(π)=d, where M(π) is a certain statistic related to the cyclic descents of π. We conjecture that b(n,d)=p(n,d) for all n and d. We prove this stronger conjecture for the cases d=1,2,3, and d=⌊(n−1)∕2⌋, and in each of these cases we establish formulas for b(n,d) involving Eulerian numbers and Eulerian–Catalan numbers.