Speaker : Prof. Yuan-Hsun Lo (National Pingtung University)
Title : Gamma-positivity for a Refinement of Median Genocchi Numbers
Time : 2021-05-14 (Fri) 15:00 - 16:00
Place : Seminar Room 638, Institute of Mathematics (NTU Campus)
Abstract: A polynomial $A(t)=a_0+a_1t+\cdots+a_nt^n$ with palindromic coefficients (i.e., $a_{n-i}=a_i$) can be written as a sum of the polynomials of the form $t^j(1+t)^{n-2j}$, \[ A(t)=\sum_{j=0}^{\lfloor n/2\rfloor} \gamma_{j}\, t^j(1+t)^{n-2j}. \] The coefficients $\gamma_j$ form a sequence called the $\gamma$-vector. The palindromic polynomial $A(t)$ is said to be $\gamma$-positive if $\gamma_j\ge 0$ for all $j$. In this talk, we shall study the generating function of descent numbers for the permutations with descent pairs of prescribed parities, the distribution of which turns out to be a refinement of median Genocchi numbers. We will prove the $\gamma$-positivity for the polynomial and derive the generating function for the $\gamma$-vectors, expressed in the form of continued fraction. We also come up with an artificial statistic that gives a $q$-analogue of the $\gamma$-positivity for the permutations with descents only allowed from an odd value to an odd value.

This is joint work with Sen-Peng Eu (NTNU), Tung-Shan Fu (NPTU) and Hsin-Hao Lai (NKNU)