Uniform Convergence of an Asymptotic Approximation to Associated Stirling Numbers

• Main Authors: Canfield, E. Rodney, Helton, J. William, Hughes, Jared A
• Format: Journal Article
• Language: English
• Published: 02-09-2024
• Subjects: Mathematics - Combinatorics
• DOI: 10.48550/arxiv.2409.01489

Let $S_r(p,q)$ be the $r$-associated Stirling numbers of the second kind, the
number of ways to partition a set of size $p$ into $q$ subsets of size at least
$r$. For $r=1$, these are the standard Stirling numbers of the second kind, and
for $r=2$, these are also known as the Ward Numbers. This paper concerns
asymptotic expansions of these Stirling numbers; such expansions have been
known for many years. However, while uniform convergence of these expansions
was conjectured in Hennecart's 1994 paper, it has not been fully proved. A
recent paper (Connamacher and Dobrosotskaya, 2020) went a long way, by proving
uniform convergence on a large set. In this paper we build on that paper and
prove convergence "everywhere."