Combinatorial and Arithmetical Properties of the Restricted and Associated Bell and Factorial Numbers

TL;DR

This paper investigates the combinatorial and arithmetical properties of generalized Bell and factorial numbers arising from restricted set partitions and permutations, providing identities, recurrences, and p-adic valuation insights.

Contribution

It introduces new combinatorial identities, recurrence relations, and p-adic valuation properties for these generalized numbers, extending classical results.

Findings

Derived new combinatorial identities and recurrences.

Analyzed p-adic valuations of the generalized numbers.

Extended classical Stirling and Bell number properties.

Abstract

Set partitions and permutations with restrictions on the size of the blocks and cycles are important combinatorial sequences. Counting these objects lead to the sequences generalizing the classical Stirling and Bell numbers. The main focus of the present article is the analysis of combinatorial and arithmetical properties of them. The results include several combinatorial identities and recurrences as well as some properties of their $p$-adic valuations.