  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.
