Invertibility of Submatrices of Pascal's Matrix and Birkhoff Interpolation

<div class="line" id="line-21"> The in&filig;nite (upper triangular) Pascal matrix is T = [ ji] for 0 &le; i, j. It is easy to see that that submatrix T (0 : n, 0 : n) is triangular with determinant 1, hence in particular, it is invertible. But what about other submatrices T (r, x) for selections r = [r0, . . . , rd] and x = [x0, . . . , xd] of the rows and columns of T ? The goal of this paper is provide a necessary and su&ffilig;cient condition for invertibility based on a connection to polynomial interpolation. In particular, we generalize the theory of Birkho&fflig; interpolation and P&uml;olya systems, and then adapt it to this problem. The result is simple: T (r, x) is invertible i&fflig; r &le; x, or equivalently, i&fflig; all diagonal entries are nonzero.</div>