The insertion encoding of restricted growth functions

TL;DR

This paper adapts insertion encodings to enumerate restricted growth functions, classifies regularity conditions, and extends results to matchings, providing new insights into their combinatorial structure.

Contribution

It introduces adapted insertion encodings for restricted growth functions and classifies when these encodings form regular languages, extending to matchings.

Findings

Classified classes with regular insertion encodings

Proved regularity conditions are the same for matchings

Extended encoding techniques to restricted growth functions

Abstract

We adapt the vertical and horizontal insertion encodings of Cayley permutations to enumerate restricted growth functions, which are in bijection with unordered set partitions. For both insertion encodings, we fully classify the classes for which these languages are regular. For the horizontal insertion encoding, we also prove that the conditions to be regular are the same for restricted growth functions of matchings.