MC-finiteness of restricted set partition functions

TL;DR

This paper investigates the property of MC-finiteness in integer sequences, especially those from set partition functions, providing methods to prove or disprove this property.

Contribution

It introduces new techniques for analyzing MC-finiteness and applies them to set partition functions, expanding understanding of their modular periodicity.

Findings

Set partition functions can be MC-finite or not, depending on their structure.

Methods for proving MC-finiteness are developed and demonstrated.

Examples show the applicability of these methods to various sequences.

Abstract

A sequence $s(n)$ of integers is MC-finite if for every $m \in \mathbb{N}^+$ the sequence $s^m(n) = s(n) \bmod{m}$ is ultimately periodic. We discuss various ways of proving and disproving MC-finiteness. Our examples are mostly taken from set partition functions, but our methods can be applied to many more integer sequences.