A Multivariate Determinant Associated with Partitions (preliminary version)

E. R. Berlekamp [1][2] raised a question concerning the entries of certain matrices of determinant 1. (Originally Berlekamp was interested only in the entries modulo 2.) Carlitz, Roselle, and Scoville [3] gave a combinatorial interpretation of the entries (over the integers, not just modulo 2) in terms of lattice paths. Here we will generalize the result of Calitz, Roselle, and Scoville in two ways: (a) we will refine the matrix entries so that they are polynomials in many indeterminates, and (b) we compute not just the determinant of these matrices, but more strongly their Smith normal form (SNF). A priori our matrices need not have a Smith normal form since they are not defined over a principal ideal domain, but the existence of SNF will follow from its explicit computation. A special case is a determinant of q-Catalan numbers. It will be more convenient for us to state our results in terms of partitions rather than lattice paths.