Counting on matrices

Abstract

In this paper, we have found formulas for the number of rectangular and symmetric matrices with the line sums divisible by a given integer. As an application, we have derived an explicit formula enumerating the number of traceless n × n, (0,1) symmetric matrices having line sums divisible by a given integer, which leads to an enumeration of labeled regular graphs with n vertices. Also, we have found a formula for the weighted enumerator (in terms of rows and columns) of rectangular matrices, which subsequently yields some nice identities satisfying curious reciprocity phenomena.