Counting compatible indexing systems for $C_{p^n}$

TL;DR

This paper counts compatible indexing systems for cyclic groups and reveals their connection to Fuss–Catalan numbers, providing recursive formulas and enumeration methods.

Contribution

It introduces a novel enumeration of compatible indexing systems for cyclic groups, linking them to Fuss–Catalan numbers and offering new recursive counting techniques.

Findings

Number of compatible indexing systems equals Fuss–Catalan numbers.

Derived recursive formulas for counting admissible sets.

Provided enumeration methods for extending indexing systems.

Abstract

We count the number of compatible pairs of indexing systems for the cyclic group $C_{p^n}$. Building on work of Balchin--Barnes--Roitzheim, we show that this sequence of natural numbers is another family of Fuss--Catalan numbers. We count this two different ways: showing how the conditions of compatibility give natural recursive formulas for the number of admissible sets and using an enumeration of ways to extend indexing systems by conceptually simpler pieces.