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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2409.01489 |
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| _version_ | 1866917767287930880 |
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| author | Canfield, E. Rodney Helton, J. William Hughes, Jared A. |
| author_facet | Canfield, E. Rodney Helton, J. William Hughes, Jared A. |
| contents | 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." |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_01489 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Uniform Convergence of an Asymptotic Approximation to Associated Stirling Numbers Canfield, E. Rodney Helton, J. William Hughes, Jared A. Combinatorics 05A16 (Primary) 05A18 (Secondary) 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." |
| title | Uniform Convergence of an Asymptotic Approximation to Associated Stirling Numbers |
| topic | Combinatorics 05A16 (Primary) 05A18 (Secondary) |
| url | https://arxiv.org/abs/2409.01489 |