Enumerative aspects of Caylerian polynomials

TL;DR

This paper explores Caylerian polynomials, which count descents in Cayley permutations, using combinatorial species and involutions to derive formulas and generating functions for these polynomials and their variants.

Contribution

It introduces new combinatorial formulas and generating functions for Caylerian polynomials, expanding understanding of permutation descent distributions.

Findings

Derived counting formulas for Caylerian polynomials

Established generating functions for Caylerian and related polynomials

Used combinatorial species and involutions in the analysis

Abstract

Eulerian polynomials record the distribution of descents over permutations. Caylerian polynomials likewise record the distribution of descents over Cayley permutations, where a Cayley permutation is a word of positive integers such that if a number appears in the word then all positive integers less than that number also appear in the word. Using combinatorial species and sign-reversing involutions we derive counting formulas and generating functions for the Caylerian polynomials as well as for related refined polynomials.