From quasisymmetric expansions to Schur expansions via a modified inverse Kostka matrix

Every symmetric function f can be written uniquely as a linear combination of Schur functions, say f=@?"@lx"@ls"@l, and also as a linear combination of fundamental quasisymmetric functions, say f=@?"@ay"@aQ"@a. For many choices of f arising in the theory of Macdonald polynomials and related areas, one knows the quasisymmetric coefficients y"@a and wishes to compute the Schur coefficients x"@l. This paper gives a general combinatorial formula expressing each x"@l as a linear combination of the y"@a's, where each coefficient in this linear combination is +1, -1, or 0. This formula arises by suitably modifying Egecioglu and Remmel's combinatorial interpretation of the inverse Kostka matrix involving special rim-hook tableaux.