q Statistics on S_n and Pattern Avoidance

2003/05/28 by Amitai Regev, Regev, Amitai, Yuval Roichman +1
Computer Science · Mathematics · #05A15 #05A19 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Bayesian Methods and Mixture Models #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:05A15 #msc:05A19

paper · pdf · doi:10.48550/arxiv.math/0305393

40 pages

arxiv created 2003/05/28 · openalex publication_date 2003/05/28 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Natural q analogues of classical statistics on the symmetric groups S_n are introduced; parameters like: the q-length, the q-inversion number, the q-descent number and the q-major index. MacMahon's theorem about the equi-distribution of the inversion number and the reverse major index is generalized to all positive integers q. It is also shown that the q-inversion number and the q-reverse major index are equi-distributed over subsets of permutations avoiding certain patterns. Natural q analogues of the Bell and the Stirling numbers are related to these q statistics -- through the counting of the above pattern-avoiding permutations.

Citations

• Generalized Pattern Avoidance
• Permutation Statistics on the Alternating Group

Related

• Permutation patterns and statistics
• Width-k Generalizations of Classical Permutation Statistics
• An extension of MacMahon's Equidistribution Theorem to ordered set partitions
• Permutation Statistics in Conjugacy Classes of the Symmetric Group
• Equidistribution of negative statistics and quotients of Coxeter groups of type B and D