Prime and Touchard Congruences of Mixed-Type Bell Numbers
Daniel Yaqubi, Madjid Mirzavaziri
Abstract
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-p2 refinements that generalize classical arithmetic properties of combinatorial sequences.
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