Real-rootedness of rook-Eulerian polynomials

TL;DR

This paper introduces rook-Eulerian polynomials, generalizes Eulerian polynomials via rook placements, proves their real-rootedness, and connects them to 312-avoiding permutations in the Bruhat order.

Contribution

It presents a new class of polynomials called rook-Eulerian polynomials, proves their real-rootedness, and links them to permutation patterns and Bruhat order.

Findings

Rook-Eulerian polynomials are real-rooted.

They generalize classical Eulerian polynomials.

Connections to 312-avoiding permutations are established.

Abstract

We introduce rook-Eulerian polynomials, a generalization of the classical Eulerian polynomials arising from complete rook placements on Ferrers boards, and prove that they are real-rooted. We show that a natural context in which to interpret these rook placements is as lower intervals of $312$-avoiding permutations in the Bruhat order. We end with some variations and generalizations along this theme.