Descents of λ-unimodal cycles in a character formula

We prove an identity conjectured by Adin and Roichman involving the descent set of λ-unimodal cyclic permutations. These permutations appear in formulas for characters of certain representations of the symmetric group. Such formulas have previously been proven algebraically. In this paper, we present a combinatorial proof for one such formula and discuss the consequences for the distribution of the descent set on cyclic permutations.