Combinatorics of higher order degenerate r-deranged bell numbers with singletons

TL;DR

This paper introduces a new generalization of barred preferential arrangements involving higher order degenerate r-deranged Bell numbers with singletons, deriving identities and asymptotic behaviors.

Contribution

It defines a novel class of combinatorial numbers extending barred preferential arrangements and provides combinatorial identities and asymptotic analysis.

Findings

Derived several combinatorial identities for the new numbers.

Provided asymptotic results for the generalized arrangements.

Abstract

When one inserts a number of identical bars in between blocks of an ordered set partition, they get a barred preferential arrangement. In this study we define a new generalization of barred preferential arrangements, by considering barred preferential arrangements with no fixed blocks, and ones where the first r elements of a set are singletons. We derive several combinatorial identities. Combinatorially these numbers are a kind of generalized barred preferential arrangements. We also provide some asymptotic results for these numbers.