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

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.