Abstract: An infinite real matrix M is said to be totally positive (TP) if every minor of M is nonnegative. Recently X. Chen, H. Liang and Y. Wang presented some sufficient conditions for the total positivity of some special lower triangular matrices.
In this talk, after reviewing some background to this field, I’ll present a combinatorial interpretation of their sufficient conditions using a classical lemma of B. Lindstrom. In particular, I’ll construct a digraph with a weight, which is positive under their sufficient conditions, such that every minor of A is equal to the sum of the weights of families of nonintersecting paths of the digraph for any Catalan-Stieltjes matrix A. An analogous result for the minors of a Hankel matrix associated to the first column of a Catalan-Stieltjes matrix will also be given. This talk is accessible to a general audience.
