On symmetric polynomials with only real zeros and nonnegative gamma vectors

The Eulerian and Narayana polynomials are the generating functions of many combinatorial objects. They appear in several areas of mathematics and share a list of common properties. In this talk I will present a family of polynomials, most of them unknown, that have the same properties as the Eulerian and Narayana polynomials. In particular, a novel and simple proof of real-rooted-ness for the Narayana polynomials is obtained as a direct consequence of our approach. Likewise, we provide an explicit formula to compute the gamma numbers for our family of polynomials. We will say a few words on the geometric and combinatorial significance for these numbers.