Séminaire Lotharingien de Combinatoire, 82B.21 (2019), 12 pp.

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

Signed Mahonian identities on permutations with subsequence restrictions

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 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.

Received: November 15, 2018. Accepted: February 17, 2019. Final version: April 1, 2019.

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