Refined sign-balance on 321-avoiding permutations

Abstract

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 2 th 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 of 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 is based on a matching method given by Elizalde and Pak.