Prime and Touchard Congruences of Mixed-Type Bell Numbers

This paper establishes a comprehensive combinatorial and arithmetic framework for mixed Stirling and mixed Bell numbers, bridging partition structures, Touchard polynomials, and prime-power congruences. Furthermore, we develop to $p$-adic valuation theory, proving prime-power Touchard congruences and higher-order modulo-$p^2$ refinements that generalize classical arithmetic properties of combinatorial sequences.