On the Graphical $r$-Stirling Numbers of the First Kind for Specific Graph Families
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866912868397481984 |
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| author | Yaqubi, Daniel Mirzavaziri, Madjid |
| author_facet | Yaqubi, Daniel Mirzavaziri, Madjid |
| contents | This paper investigates the \textbf{graphical $r$-Stirling numbers of the first kind}, denoted by $\str{G}{k}$, which enumerate partitions of a vertex set $V(G)$ into $k$ disjoint cycles such that $r$ specified vertices occupy distinct blocks. We establish closed-form expressions and recursive identities for fundamental graph families, including \textbf{Path} ($P_n$), \textbf{Cycle} ($C_n$), \textbf{Star} ($S_n$), \textbf{Wheel} ($W_n$), and \textbf{Fan} ($F_n$) graphs.
A primary focus of this study is the \textbf{statistical characterization} of the cycle distribution. We derive explicit formulas for the \textbf{mean} and \textbf{variance} of these numbers, extracted from the structural properties of the $r$-cycle polynomials. These results provide a rigorous measure of the average cycle density and variability across different graph topologies, bridging the gap between algebraic combinatorics and the structural analysis of restricted permutations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_02046 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the Graphical $r$-Stirling Numbers of the First Kind for Specific Graph Families Yaqubi, Daniel Mirzavaziri, Madjid Combinatorics This paper investigates the \textbf{graphical $r$-Stirling numbers of the first kind}, denoted by $\str{G}{k}$, which enumerate partitions of a vertex set $V(G)$ into $k$ disjoint cycles such that $r$ specified vertices occupy distinct blocks. We establish closed-form expressions and recursive identities for fundamental graph families, including \textbf{Path} ($P_n$), \textbf{Cycle} ($C_n$), \textbf{Star} ($S_n$), \textbf{Wheel} ($W_n$), and \textbf{Fan} ($F_n$) graphs. A primary focus of this study is the \textbf{statistical characterization} of the cycle distribution. We derive explicit formulas for the \textbf{mean} and \textbf{variance} of these numbers, extracted from the structural properties of the $r$-cycle polynomials. These results provide a rigorous measure of the average cycle density and variability across different graph topologies, bridging the gap between algebraic combinatorics and the structural analysis of restricted permutations. |
| title | On the Graphical $r$-Stirling Numbers of the First Kind for Specific Graph Families |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2602.02046 |