Total non-negativity of some combinatorial matrices

Abstract

Many combinatorial matrices — such as those of binomial coefficients, Stirling numbers of both kinds, and Lah numbers — are known to be totally non-negative, meaning that all minors (determinants of square submatrices) are non-negative.The examples noted above can be placed in a common framework: for each one there is a non-decreasing sequence (a1,a2,…), and a sequence (e1,e2,…), such that the (m,k) entry of the matrix 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).We consider this general framework. For a non-decreasing sequence (a1,a2,…) we establish necessary and sufficient conditions on the sequence (e1,e2,…) for the corresponding matrix to be totally non-negative. As corollaries we obtain total non-negativity of matrices of rook numbers of Ferrers boards, and of graph Stirling numbers of chordal graphs.