On the periodicity problem of residual r-Fubini sequences

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.