Permutation Statistics on the Alternating Group

2003/02/25 by Amitai Regev, Regev, Amitai, Yuval Roichman +1 · 1 citation
Mathematics · #05A15 #05A19 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:05A15 #msc:05A19

paper · pdf · doi:10.48550/arxiv.math/0302301

45 pages

arxiv created 2003/02/25 · openalex publication_date 2003/02/25 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let A_n⊆ S_n denote the alternating and the symmetric groups on 1,...,n. MacMahaon's theorem, about the equi-distribution of the length and the major indices in S_n, has received far reaching refinements and generalizations, by Foata, Carlitz, Foata-Schutzenberger, Garsia-Gessel and followers. Our main goal is to find analogous statistics and identities for the alternating group A_n. A new statistic for S_n, \it the delent number, is introduced. This new statistic is involved with new S_n equi-distribution identities, refining some of the results of Foata-Schutzenberger and Garsia-Gessel. By a certain covering map f:A_n+1→ S_n, such S_n identities are `lifted' to A_n+1, yielding the corresponding A_n+1 equi-distribution identities.

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• q Statistics on S_n and Pattern Avoidance

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