Signed mahonian identities on permutations with subsequence restrictions

Sen Peng Eu, Tung Shan Fu, Hsiang Chun Hsu, Hsin Chieh Liao, Wei Liang Sun

Research output: Contribution to journalArticle

Abstract

In this paper, we present a number of results surrounding Caselli's conjecture on the equidistribution of the major index with sign over the two subsets of permutations of {1,2,…,n} containing respectively the word 12⋯k and the word (n−k+1)⋯n as a subsequence, under a parity condition of n and k. We derive broader bijective results on permutations containing varied subsequences. As a consequence, we obtain the signed mahonian identities on families of restricted permutations, in the spirit of a well-known formula of Gessel–Simion, covering a combinatorial proof of Caselli's conjecture. We also derive an extension of the insertion lemma of Han and Haglund–Loehr–Remmel which allows us to obtain a signed enumerator of the major-index increments resulting from the insertion of a pair of consecutive numbers in any place of a given permutation.

Original languageEnglish
Article number105131
JournalJournal of Combinatorial Theory. Series A
Volume170
DOIs
Publication statusPublished - 2020 Feb
Externally publishedYes

Keywords

  • Equidistribution
  • Insertion lemma
  • Linear extensions
  • Pattern avoiding permutations
  • Permutations with subsequence restrictions
  • Signed major index

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics

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