The deranged Bell numbers

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.