Signed Mahonian Identities on Permutations with Subsequence Restrictions

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.