Slicing Skew-tableau Frames

Abstract

Using special decompositions of the frame into zigzag paths, a simple algorithm is given for reducing the enumeration of skew-tableaux to the enumeration of k -tuples of linear partitions. This gives a direct combinatorial method for obtaining the homomorphic image of a frame in the ring Z [ x 1 x 2 , ...] of polynomials in an infinite sequence of independent indeterminates x i and thus gives a combinatorial interpretation to the non-zero terms that arise from the classical expression for a skew Schur function as a determinant of homogeneous symmetric functions (or of elementary symmetric functions). In the special case of tableau frames, it provides direct combinatorial significance to the signed Kostka numbers of the second kind. Also, skew-tableaux are generalized by defining arena maps, i.e. row monotonic mappings that can fail to be strictly increasing on columns only in specified ways, and the algorithm is extended to include their enumeration.