A combinatorial rule for the schur function expansion of the PlethysmS(1a,b)[Pk

Abstract

We present a simple combinatorial rule to expand the plethysm Pn [S(1 a ,b)](x) of a power summetric function Pn (x) and a Schur function of hook shapeS(1 a ,b)(x), as a sum of Schur functions. The key ingredient of our proof is a correspondence between the circle diagrams, introduced by Chen, Garsia and Remmel [6] in their SXP algorithm to compute the Schur function expansion of Pn [S λ], and certain special rim hook and transposed special rim hook tabloids which is of interest in its own right. As an application of our rule, we drive explicit formulas for the coefficient of any Schur function of hook shape in the Schur function expansion of a plethysm of any two Schur functions of hook shape.