Smith Normal Form of a Multivariate Matrix Associated with Partitions
نویسندگان
چکیده
Abstract. Considering a question of E. R. Berlekamp, Carlitz, Roselle, and Scoville gave a combinatorial interpretation of the entries of certain matrices of determinant 1 in terms of lattice paths. Here we generalize this result by refining the matrix entries to be multivariate polynomials, and by determining not only the determinant but also the Smith normal form of these matrices. A priori the Smith form need not exist but its existence follows from the explicit computation. It will be more convenient for us to state our results in terms of partitions rather than lattice paths.
منابع مشابه
Smith Normal Form of a Multivariate Matrix 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 Carlitz, Roselle, and Scoville in two...
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