Arithmetic Properties of Mixed Stirling Numbers of the second kind

We investigate the mixed Stirling numbers of the second kind, $\mathcal{S}(n; \mathbf{c})$, which count partitions of $n$ distinct elements into $m$ unlabeled and $k$ labeled non-empty blocks encoded by $\mathbf{c} = (m, 1^k)$. We establish their recurrence relations and exponential generating functions, and analyze their behavior modulo a prime $p$ and $p^2$. In particular, we extend the classical Touchard congruence to this mixed framework. Our results show that these configurations possess unique number-theoretic signatures distinct from classical set partitions.