The insertion encoding of Cayley permutations

TL;DR

This paper introduces new insertion encodings for Cayley permutations, classifies when these languages are regular, and provides algorithms for enumeration, solving an open problem in permutation class enumeration.

Contribution

It generalizes insertion encoding to Cayley permutations, classifies regularity of these languages, and offers algorithms for enumeration, including solving an open problem.

Findings

Classified Cayley permutation classes with regular insertion encoding languages.

Developed an algorithm to compute rational generating functions.

Enumerated hare pop-stack sortable Cayley permutations, solving an open problem.

Abstract

We introduce the vertical and horizontal insertion encodings for Cayley permutations which naturally generalise the insertion encoding for permutations. In both cases, we fully classify the Cayley permutation classes for which these languages are regular, and provide an algorithm for computing the rational generating functions. We use our algorithm to solve an open problem of Cerbai by enumerating the hare pop-stack sortable Cayley permutations.