# Real-rootedness of variations of Eulerian polynomials

## TL;DR

This paper proves the real-rootedness of generalized Eulerian polynomials related to $ extbf{s}$-inversion sequences, confirming conjectures and revealing geometric interpretations, thus advancing understanding of their algebraic and combinatorial properties.

## Contribution

It generalizes Eulerian polynomials to $ extbf{s}$-inversion sequences and proves their real-rootedness using interlacing polynomials, confirming conjectures and connecting to geometric structures.

## Key findings

- Proved real-rootedness of generalized Eulerian polynomials.
- Confirmed conjecture on binomial Eulerian polynomials.
- Established geometric interpretation via edgewise subdivisions.

## Abstract

The binomial Eulerian polynomials, introduced by Postnikov, Reiner, and Williams, are $\gamma$-positive polynomials and can be interpreted as $h$-polynomials of certain flag simplicial polytopes. Recently, Athanasiadis studied analogs of these polynomials for colored permutations. In this paper, we generalize them to $\mathbf{s}$-inversion sequences and prove that these new polynomials have only real roots by the method of interlacing polynomials. Three applications of this result are presented. The first one is to prove the real-rootedness of binomial Eulerian polynomials, which confirms a conjecture of Ma, Ma, and Yeh. The second one is to prove that the symmetric decomposition of binomial Eulerian polynomials for colored permutations is real-rooted. Thirdly, our polynomials for certain $\mathbf{s}$-inversion sequences are shown to admit a similar geometric interpretation related to edgewise subdivisions of simplexes.