Coefficientwise Total Positivity of Hankel Matrices in Counting Problems

A matrix M with real entries is said to be totally positive (TP) if all its square submatrices have non-negative determinants. We can extend this definition to matrices with polynomial entries where a polynomial is said to be coefficientwise non-negative if all coefficients are non-negative. This talk will begin with a quick introduction to total positivity. We will then introduce Hankel matrices and provide a characterisation of totally positive Hankel matrices due to Gantmakher and Krein and show its relation to the Stieltjes moment problem and to continued fractions. Next, we will give examples of sequences occurring in combinatorics whose Hankel matrices are totally positive. We then generalise these notions to coefficientwise total positivity. We will conclude this talk by mentioning some of the proof techniques in this area. If time permits, we will mention some of our recent results. No prerequisites will be necessary to follow the talk.