Combinatorics Seminar

16:00-17:00

Speaker: Han Bin

Title: Some multivariate polynomials for doubled permutations

Abstract: Flajolet and Françon [European. J. Combin. 10 (1989) 235-241] gave a combinatorial interpretation for the Taylor coefficients of the Jacobian elliptic functions in terms of doubled permutations. We show that a multivariable counting of the doubled permutations has also an explicit continued fraction expansion generalizing the continued fraction expansions of Rogers and Stieltjes.

The second goal of this talk is to study the expansion of the Taylor coefficients of the generalized Jacobian elliptic functions, which implies the symmetric and unimodal property of the Taylor coefficients of the generalized Jacobian elliptic functions.

17:00-18:00

Speaker: Marius Tiba

Title: Sharp stability of Brunn-Minkowski for homothetic regions

Abstract: We prove a sharp stability result concerning how close homothetic sets attaining near-equality in the Brunn-Minkowski inequality are to being convex. In particular, resolving a conjecture of Figalli and Jerison, we show there are universal constants $C_n,d_n>0$ such that for $A \subset \mathbb{R}^n$ of positive measure, if $|\frac{A+A}{2}\setminus A| \le d_n |A|$, then $|\text{conv}(A)\setminus A| \le C_n |\frac{A+A}{2}\setminus A|$ for $\text{conv}(A)$ the convex hull of $A$.

Joint work with Peter van Hintum and Hunter Spink