The deranged Bell numbers

TL;DR

This paper introduces deranged Bell numbers counting deranged partitions of a set, exploring their properties, asymptotic behavior, and variants, thus extending classical Bell number theory.

Contribution

It presents the first study of deranged Bell numbers, including formulas, asymptotics, and generalizations, expanding the combinatorial understanding of set partitions.

Findings

Deranged Bell numbers have explicit formulas and generating functions.

Asymptotic behavior of deranged Bell numbers is characterized.

Initial results on r-deranged Bell numbers are provided.

Abstract

It is known that the ordered Bell numbers count all the ordered partitions of the set $[n]=\{1,2,\dots,n\}$. In this paper, we introduce the deranged Bell numbers that count the total number of deranged partitions of $[n]$. We first study the classical properties of these numbers (generating function, explicit formula, convolutions, etc.), we then present an asymptotic behavior of the deranged Bell numbers. Finally, we give some brief results for their $r$-versions.