The sorting index and set-valued joint equidistributions of B_n and D_n

2014/03/10 by Sen‐Peng Eu, Eu, Sen-Peng, Yuan–Hsun Lo +3
Computer Science · Mathematics · #05A05 #05A19 #Advanced Combinatorial Mathematics #Algorithms and Data Compression #Combinatorics (math.CO) #FOS: Mathematics #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.1403.2169

openalex publication_date 2014/03/10 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

The sorting indices sor_B and sor_D on the Coxeter groups of type B and D respectively are defined by Petersen and it is proved that (inv_B, rlmin) and (sor_B, ℓ'_B) have the same joint distribution for type B while inv_D and sor_D have the same distribution for type D. These results, including a set-valued extension of type B involving two equildistributed pairs of three statistics, are proved combinatorially by Chen et al. via two mappings φ:=(B-code)^-1∘ (A-code) and ψ:=(D-code)^-1∘ (C-code). In this paper we further extend these results. In type B we prove a set-valued joint equildistribution between a pair of seven statistics, and find a five-variable generating function. In type D we define new set-valued statistics, among them Cyc⁺_D and Cyc^-_D, and firstly find a set-valued joint equidistribution between a pair of five statistics and find a four-variable generating function.

Related

• Set-valued sorting index and joint equidistributions
• The Sorting Index and Permutation Codes
• The sorting index on colored permutations and even-signed permutations
• Equidistribution of negative statistics and quotients of Coxeter groups of type B and D
• On some Euler-Mahonian distributions