The $r$-derangement numbers

TL;DR

This paper explores generalized derangement numbers, providing exact formulas, combinatorial relations, and analyzing their number-theoretic properties, including modularity and p-adic valuations.

Contribution

It introduces a comprehensive study of generalized derangements, extending classical enumeration with new formulas, relations, and number-theoretic insights.

Findings

Derived exact formulas for generalized derangements

Established combinatorial relations and asymptotic behavior

Analyzed number-theoretic properties like modularity and p-adic valuations

Abstract

The classical derangement numbers count fixed point-free permutations. In this paper we study the enumeration problem of generalized derangements, when some of the elements are restricted to be in distinct cycles in the cycle decomposition. We find exact formula, combinatorial relations for these numbers as well as analytic and asymptotic description. Moreover, we study deeper number theoretical properties, like modularity, $p$-adic valuations, and diophantine problems.