Pattern-avoiding Cayley permutations via combinatorial species

TL;DR

This paper studies pattern-avoiding Cayley permutations using combinatorial species, deriving formulas and exploring equivalences, thus advancing understanding of their structure and enumeration.

Contribution

It introduces species equations and counting formulas for pattern-avoiding Cayley permutations and generalizes to primitive structures, providing new tools and insights.

Findings

Derived species equations and generating series for pattern-avoiding Cayley permutations

Established counting formulas for patterns of length up to three

Explored Wilf equivalence notions in this context

Abstract

A Cayley permutation is a word of positive integers such that if a letter appears in this word, then all positive integers smaller than that letter also appear. We initiate a systematic study of pattern avoidance on Cayley permutations adopting a combinatorial species approach. Our methods lead to species equations, generating series, and counting formulas for Cayley permutations avoiding any pattern of length at most three. We also introduce the species of primitive structures as a generalization of Cayley permutations with no "flat steps". Finally, we explore various notions of Wilf equivalence arising in this context.