Degenerate r-associated Stirling numbers

TL;DR

This paper introduces degenerate r-associated Stirling numbers of both kinds, extending classical definitions and deriving recurrence relations, thereby enriching combinatorial enumeration methods.

Contribution

It defines degenerate versions of r-associated Stirling numbers and establishes their recurrence relations, expanding the combinatorial toolkit.

Findings

Derived recurrence relations for degenerate r-associated Stirling numbers.

Showed these numbers reduce to classical degenerate Stirling numbers when r=1.

Extended the combinatorial framework for set partitions with minimum subset sizes.

Abstract

For any positive integer r, the r-associated Stirling number of the second kind enumerates the number of partitions of the set{1,2,3,...,n} into k non-empty disjoint subsets such that each subset contains at least r elements. We introduce the degenerate r-associated Stirling numbers of the second kind and of the first kind. They are degenerate versions of the r-associated Stirling numbers of the second kind and of the first kind, and reduce to the degenerate Stirling numbers of the second kind and of the first kind for r=1. The aim of this paper is to derive recurrence relations for both of those numbers.