Refined sign-balance on 321-avoiding permutations

Details Refined sign-balance on 321-avoiding permutations Refined sign-balance on 321-avoiding permutations

2003-05-23

arxiv.org/abs/math/0305327v1

Mathematics

Combinatorics

11 pages

Scientific paper

The number of even 321-avoiding permutations of length n is equal to the number of odd ones if n is even, and exceeds it by the (n-1)/2th Catalan number otherwise. We present an involution that proves a refinement of this sign-balance property respecting the length of the longest increasing subsequence of the permutation. In addition, this yields a combinatorial proof of a recent analogous result of Adin and Roichman dealing with the last descent. In particular, we answer the question how to obtain the sign of a 321-avoiding permutation from the pair of tableaux resulting from the Robinson-Schensted-Knuth algorithm. The proof of the simple solution bases on a matching method given by Elizalde and Pak.

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