Inversion generating functions for signed pattern avoiding permutations

Department(s)

Natural Science

Document Type

Article

Publication Date

2-17-2017

Keywords

Generating function, Inversion statistic, Major index, Pattern avoiding permutations, Signed permutations

Abstract

We consider the classical Mahonian statistics on the set B (Σ) of signed per- mutations in the hyperoctahedral group B which avoid all patterns in Σ, where Σ is a set of patterns of length two. In 2000, Simion gave the cardinality of B (Σ) in the cases where Σ contains either one or two patterns of length two and showed that |B (Σ]| is constant whenever |Σ| = 1, whereas in most but not all instances where |Σ| = 2, |B (Σ)| = (n + 1)!. We answer an open question of Simion by providing bijections from B (Σ) to S in these cases where |B (Σ)| = (n + 1)!. In addition, we extend Simion’s work by providing a combinatorial proof in the language of signed permutations for the major index on B (21,21) and by giving the major index on D (Σ) for Σ = {21,21} and Σ = {21,21}. The main result of this paper is to give the inversion generating functions for B (Σ) for almost all sets Σ with |Σ| ≤ 2. n n n n n n n+1 n n n n

Publication Title

Electronic Journal of Combinatorics

E-ISSN

10778926

Volume

24

Issue

1

DOI

10.37236/6545

Link to Full Text

Share

COinS