Combinatorics of the immaculate inverse Kostka matrix

Abstract

The classical Kostka matrix counts semistandard tableaux and expands Schur symmetric functions in terms of monomial symmetric functions. The entries in the inverse Kostka matrix can be computed by various algebraic and combinatorial formulas involving determinants, special rim hook tableaux, raising operators, and tournaments. Our goal here is to develop an analogous combinatorial theory for the inverse of the immaculate Kostka matrix. The immaculate Kostka matrix enumerates dual immaculate tableaux and gives a combinatorial definition of the dual immaculate quasisymmetric functions Sα. We develop several formulas for the entries in the inverse of this matrix based on suitably generalized raising operators, tournaments, and special rim-hook tableaux. Our analysis reveals how the combinatorial conditions defining dual immaculate tableaux arise naturally from algebraic properties of raising operators. We also obtain an elementary combinatorial proof that the definition of Sα via dual immaculate tableaux is equivalent to the definition of the immaculate noncommutative symmetric functions Sα via noncommutative Jacobi–Trudi determinants. A factorization of raising operators leads to bases of NSym interpolating between the S-basis and the h-basis, and bases of QSym interpolating between the S∗-basis and the M -basis. We also give t-analogues for most of these results using combinatorial statistics defined on dual immaculate tableaux and tournaments.