Caylerian polynomials

TL;DR

This paper introduces Caylerian polynomials, extending Eulerian polynomials to Cayley permutations, and explores their properties and connections to Burge structures, revealing new combinatorial insights.

Contribution

It generalizes classical descent polynomial results to Cayley permutations and links them to Burge words and matrices, providing new combinatorial frameworks.

Findings

Caylerian polynomials are linked to Burge words and matrices.

The $b3$-nonnegativity of two-sided Eulerian polynomials is reformulated.

Cayley permutations with fixed ascent sets are counted by Burge matrices with fixed row sums.

Abstract

The Eulerian polynomials enumerate permutations according to their number of descents. We initiate the study of descent polynomials over Cayley permutations, which we call Caylerian polynomials. Some classical results are generalized by linking Caylerian polynomials to Burge words and Burge matrices. The $\gamma$-nonegativity of the two-sided Eulerian polynomials is reformulated in terms of Burge structures. Finally, Cayley permutations with a prescribed ascent set are shown to be counted by Burge matrices with fixed row sums.