Richard P. Stanley Seminar in Combinatorics: Coefficientwise Hankel-total positivity

SEMINARS, HARVARD-MIT COMBINATORICS

Speaker:

Alan Sokal - University College London

A matrix M of real numbers is called totally positiveif every minor of $M$ is nonnegative. Gantmakher and Krein showedin 1937 that a Hankel matrix $H = (a_{i+j})_{i,j \ge 0}$of real numbers is totally positive if and only if the underlyingsequence $(a_n)_{n \ge 0}$ is a Stieltjes moment sequence,i.e.\ the moments of a positive measure on $[0,\infty)$.Here I will introduce a generalization: a matrix $M$ of polynomials(in some set of indeterminates) will be called{\em coefficientwise totally positive}\/ if every minor of $M$is a polynomial with nonnegative coefficients. And a sequence$(a_n)_{n \ge 0}$ of polynomials will be called{\em coefficientwise Hankel-totally positive}\/ if the Hankel matrix$H = (a_{i+j})_{i,j \ge 0}$ associated to $(a_n)$ is coefficientwisetotally positive.It turns out that many sequences of polynomials arising naturallyin enumerative combinatorics are (empirically) coefficientwiseHankel-totally positive. I will discuss some methods for proving this.But these proofs fall far short of what appears to be true.

===============================

For more info, see https://math.mit.edu/combin/