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 (a_1, a_2, ...), and a sequence (e_1, e_2, ...), such that the (m,k)-entry of the matrix is the coefficient of the polynomial (x-a_1)...(x-a_k) in the expansion of (x-e_1)...(x-e_m) as a linear combination of the polynomials 1, x-a_1, ..., (x-a_1)...(x-a_m).

We consider this general framework. For a non-decreasing sequence a_1, a_2, ... we establish necessary and sufficient conditions on the sequence e_1, e_2, ... for the corresponding matrix to be totally non-negative.

This is joint work with Adrian Pacurar, Notre Dame. Speaker(s): David Galvin (University of Notre Dame)