16/09/2016, 11:20 — 11:45 — Room 6.2.38, Faculty of Sciences of the Universidade de Lisboa
José Agapito Ruiz, Universidade de Lisboa
$\gamma$-numbers
Given a polynomial $f(t)=a_0+a_1 t+\cdots+a_n t \in\mathbb{R}[t]$, the $\gamma$-numbers of $f$ are the coefficients of this polynomial in its expansion with respect to the basis
\[ \Big\{ (1+t)^n, t(1+t)^{n-2},\ldots,t^{\lfloor \frac{n}{2} \rfloor} (1+t)^{n-2\lfloor \frac{n}{2} \rfloor},t^{\lfloor \frac{n}{2} \rfloor+1},\ldots,t^n\Big\}.\] In particular, if $f$ is palindromic (symmetric), its $\gamma$-numbers may be positive, negative, or zero (some of these numbers are necessarily zero). The $\gamma$-numbers are especially interesting when they are positive integers, since they can be associated to the counting of various combinatorial objects. The standard Eulerian and Narayana polynomials are two well-known examples of palindromic polynomials whose $\gamma$-numbers are positive integers.
The purpose of this talk is to present a general formula to compute the $\gamma$-numbers of any polynomial. We will pay special attention to the $\gamma$-numbers of a class of polynomials that contains the Eulerian and the Narayana polynomials, and discuss some of their combinatorial and geometrical interpretations.
