Speaker:  Alan Sokai  (University College London)

Title:  Coefficientwise Hankel-total positiviity

Abstract;

A matrix M of real numbers is called totally positive if every minor of M is nonnegative.  Gantmakher and Krein showed in 1937 that a Hankel matrix of real numbers is totally positive if and only if the underlying sequence is a Stieltjes moment sequence, i.e. the moments of a positive measure on the nonnegative real axis.

Here I will introduce a generalization:  a matrix M of polynomials (in some set of indeterminates) will be called coefficientwise totally positive if every minor of M is a polynomial with nonnegative coefficients.   And a sequence of polynomials will be called coefficientwise Hankel-totally positive if the Hankel matrix associated to that sequence is coefficientwise totally positive.It turns out that many sequences of polynomials arising naturally in enumerative combinatorics are (empirically) coefficientwise Hankel-totally positive.  I will discuss some methods for proving this. But these proofs fall far short of what appears to be true.