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.

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