Totally non-negativity of a family of change-of-basis matrices

Abstract

Let a=(a1,a2,…,an) and e=(e1,e2,…,en) be real sequences. Denote by Me→a the (n+1)×(n+1) matrix whose (m,k) entry (m,k∈{0,…,n}) is the coefficient of the polynomial (x−a1)⋯(x−ak) in the expansion of (x−e1)⋯(x−em) as a linear combination of the polynomials 1,x−a1,…,(x−a1)⋯(x−am). By appropriate choice of a and e the matrix Me→a can encode many familiar doubly-indexed combinatorial sequences, such as binomial coefficients, Stirling numbers of both kinds, Lah numbers and central factorial numbers.In all four of these examples, Me→a enjoys the property of total non-negativity — the determinants of all its square submatrices are non-negative. This leads to a natural question: when, in general, is Me→a totally non-negative?Galvin and Pacurar found a simple condition on e that characterizes total non-negativity of Me→a when a is non-decreasing. Here we fully extend this result. For arbitrary real sequences a and e, we give a condition that can be checked in O(n2) time that determines whether Me→a is totally non-negative. When Me→a is totally non-negative, we witness this with a planar network whose weights are non-negative and whose path matrix is Me→a. When it is not, we witness this with an explicit negative minor.