On the $r$-Derangements of type B

TL;DR

This paper explores the combinatorial properties and generating functions of certain bounded set partitions with coloring constraints, establishing connections with Riordan arrays.

Contribution

It introduces new combinatorial structures related to $r$-derangements of type B and analyzes their properties and generating functions.

Findings

Derived new formulas for counting these partitions.

Established links between these partitions and Riordan arrays.

Provided combinatorial interpretations of the generating functions.

Abstract

Extensions of a set partition obtained by imposing bounds on the size of the parts and the coloring of some of the elements are examined. Combinatorial properties and the generating functions of some counting sequences associated with these partitions are established. Connections with Riordan arrays are presented.