A combinatorial interpretation of the inverse t-Kostka matrix

In this paper we use tournament matrices to give a combinatorial interpretation for the entries of the inverse t-Kostka matrix, which is the transition matrix between the Hall-Littlewood polyno-mials and the Schur functions. 0.1 Introduction In the rst section of this paper we introduce some basic notation about tournament matrices, and prove a theorem that is crucial in the second section. In the second section we prove that the entries of the inverse t-Kostka matrix can be interpreted combinatorially as a weighted sum over a subset of tournament matrices, which we call Special Tournament Matrices. We refer the reader to ((Mac]) for an introduction to Symmetric functions and their q-analogs. Then in the third section we relate Special Rim Hook Tabloids, which were introduced by Egecioglu and Remmel to interpret the inverse Kostka matrix (see E-R1]), and Special Tournament Matrices. In the fourth section we exploit the combinatorial deenition just developed to obtain more direct ways to obtain some entries of the inverse t-Kostka matrix. Appendix A surveys the connection among bubble diagrams (developed by Chen-Garsia-Remmel to describe the plethysm S n S m ]) (see C-G-R]), Special Rim Hook Tabloids, and Special Tournament Matrices. Appendix B lists some conjectures and tables of values for the inverse t-Kostka matrix. In this section we shall state some preliminary results about tournament matrices. The reader is referred to ((Moon]) for a more complete introduction to Tournament Matrices. Deenition 1 A Tournament Matrix is a n n matrix (m ij) with entries from the set f0,1g such that the entries in the main diagonal are all 0 ! , 0 1 1 0 0 0 0 1 0 ! , 0 1 0 0 0 1 1 0 0 ! , 0 1 0 0 0 0 1 1 0 ! , 0 0 1 1 0 1 0 0 0 ! , 0 0 1 1 0 0 0 1 0 ! , 0 0 0 1 0 1 1 0 0 ! , 0 0 0 1 0 0 1 1 0 !. Tournament Matrices of size n (we'll denote this set ? n) will be used to describe the transition matrix between the set of Hall-Littlewood polynomials fP : ` ng and the set of Schur functions fs : ` ng, two ordered bases for (t) (see Mac]). This point of view gives rise to some exiting combinatorial questions, some …