Invertibility of Submatrices of the Pascal Matrix and Birkhoff Interpolation

The infinite upper triangular Pascal matrix is T = [( j )i] for 0 ≤ i, j. It is easy to see that any leading principle square submatrix is triangular with determinant 1, hence invertible. In this paper, we investigate the invertibility of arbitrary square submatrices Tr, c comprised of rows r = [r0, … , rm ] and columns c = c0 , … , cm[] of T. We show that Tr, c is invertible r ≤ c i.e., ri ≤ ci for i = 0, …, m(), or equivalently, iff all diagonal entries are nonzero. To prove this result, we establish a connection between the invertibility of these submatrices and polynomial interpolation. In particular, we apply the theory of Birkhoff interpolation and Pölya systems.