Séminaire Lotharingien de Combinatoire, 93B.47 (2025), 12 pp.
Kevin Liu and Mei Yin
Descents and Flag major Index on Conjugacy Classes of Colored Permutation Groups without Short Cycles
Abstract.
We consider the descent and flag major index statistics on the colored permutation groups Sn,r = Zr ≀ Sn and their conjugacy classes. We show that the k-th moments of these statistics on Sn,r will coincide with the corresponding moments on all conjugacy classes with no cycles of lengths 1,2,...,2k. Using this, we establish the asymptotic normality of the descent and flag major index statistics on conjugacy classes of Sn,r with sufficiently long cycles. Our results generalize prior work of Fulman involving the descent and major index statistics on the symmetric group Sn. Our methods involve an intricate extension of Fulman's work on Sn combined with the theory of degrees for colored permutation statistics, as introduced by Campion Loth, Levet, Liu, Sundaram, and Yin.
Received: November 15, 2024.
Accepted: February 15, 2025.
Final version: April 1, 2025.
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