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

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