Power-product matrix: nonsingularity, sparsity and determinant

In this paper, we are interested in a special class of integer matrices, namely the power-product matrix, defined with two positive integers n and d. Each matrix element is computed by weak compositions d into parts. The has several interesting applications such as power-sum representation polynomials difference-of-convex-sums-of-squares decomposition polynomials. We investigate some properties including: nonsingularity, sparsity determinant. Based on techniques enumerative combinatorics, prove that nonsingular number nonzero entries can be exactly. This shows sparse structure which good feature numerical computation its inverse required applications. Special attention devoted to determinant for = 2 whose explicit formulation obtained.