Sign-balances of Tableaux with at Most Three Rows

Abstract

Let the sign of a standard Young tableau (SYT) be the sign of the permutation obtained by reading the entries row by row from left to right, starting from the top row. By a sign-reversing involution on the tableaux in terms of lattice paths, we obtain the sign-balance of $n$-cell SYTs of all shapes with at most three rows. For skew shapes, we obtain partial results on the sign-balance enumeration of $2n$-cell skew SYTs of all shapes with at most three rows, under a parity condition of the skew part.