# Prolific Compositions

## TL;DR

This paper investigates the property of prolificity in combinatorial structures, focusing on integer compositions, and characterizes when compositions are prolific for given patterns, providing automata-based recognition and classification of minimal cases.

## Contribution

It introduces the concept of prolificity in integer compositions, characterizes prolific patterns, and constructs automata to recognize prolific compositions, including classification of minimal instances.

## Key findings

- Prolific compositions occur if and only if the pattern starts and ends with 1.
- Automata can be constructed to recognize prolific compositions for each pattern.
- Some patterns have a unique minimal prolific composition, which are classified.

## Abstract

Under what circumstances might every extension of a combinatorial structure contain more copies of another one than the original did? This property, which we call prolificity, holds universally in some cases (e.g., finite linear orders) and only trivially in others (e.g., permutations). Integer compositions, or equivalently layered permutations, provide a middle ground. In that setting, there are prolific compositions for a given pattern if and only if that pattern begins and ends with 1. For each pattern, there is an easily constructed automaton that recognises prolific compositions for that pattern. Some instances where there is a unique minimal prolific composition for a pattern are classified.