Signed Mahonian Identities on Permutations with Subsequence Restrictions

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

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

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 f1, 2,…, ng containing respectively the word 1 2…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 and Simion, covering a combinatorial proof of Caselli’s conjecture. We also derive an extension of the insertion lemma of Haglund, Loehr, and 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 number#21
JournalSeminaire Lotharingien de Combinatoire
Issue number82
Publication statusPublished - 2019

Keywords

  • Signed mahonian statistics
  • major index with sign
  • subsequence restrictions

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics

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