## Total non-negativity of some combinatorial matrices

- Feb. 21, 2020
- 2:30 p.m.

## 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.

These examples can be put in a common framework: for each there is a non-decreasing sequence $(a _ 1, a _ 2, \ldots)$, and a sequence $(e _ 1, e _ 2, \ldots)$, such that the $(m,k)$-entry of the matrix is the coefficient of the polynomial $(x-a _ 1)\cdots(x-a _ k)$ in the expansion of $(x-e _ 1)\cdots(x-e _ m)$ as a linear combination of the polynomials $1, x-a _ 1, \ldots, (x-a _ 1)\cdots(x-a _ m)$.

I’ll consider this general framework. For a non-decreasing sequence $(a _ 1, a _ 2, \ldots)$ I’ll give necessary and sufficient conditions on the sequence $(e _ 1, e _ 2, \ldots)$ for the corresponding matrix to be totally non-negative.

This is joint work with Adrian Pacurar, Notre Dame.