The permutations of horse racing, where ties are possible, are counted by the $\textit{Fubini numbers}$, also called the $\textit{horse numbers}$. The $\textit{r-horse numbers}$ are a counting of such horse race finishes where some subset of $r$ horses agree to finish the race in a specific strong ordering. The $r$-horse numbers for fixed $r$ are expressed as a sum of $r$ index shifted sequences of Fubini numbers weighted with the $\textit{signed Stirling numbers of the first kind}$. Then eventual modular periodicity of $\textit{r-Fubini numbers}$ is shown and their maximum period is determined to be the $\textit{Carmichael function}$ of the modulus. The maximum period is attained in the case of an odd modulus for Fubini numbers.