On the periodicity problem of residual r-Fubini sequences

TL;DR

This paper investigates the periodicity of the sequence of $r$-Fubini numbers modulo an integer, proving its periodic nature and providing a method to compute the period, along with an explicit formula for $r$-Stirling numbers.

Contribution

The paper establishes the periodicity of the $r$-Fubini sequence modulo any positive integer and derives a formula for its period, also providing an explicit expression for $r$-Stirling numbers.

Findings

The $r$-Fubini sequence modulo $s$ is periodic.

A method to calculate the period length of the sequence.

An explicit formula for $r$-Stirling numbers.

Abstract

For any positive integer $r$, the $r$-Fubini number with parameter $n$, denoted by $F_{n,r}$, is equal to the number of ways that the elements of a set with $n+r$ elements can be weak ordered such that the $r$ least elements are in distinct orders. In this article we focus on the sequence of residues of the $r$-Fubini numbers modulo a positive integer $s$ and show that this sequence is periodic and then, exhibit how to calculate its period length. As an extra result, an explicit formula for the $r$-Stirling numbers is obtained which is frequently used in calculations.