On the Graphical $r$-Stirling Numbers of the First Kind for Specific Graph Families

TL;DR

This paper introduces and analyzes the graphical $r$-Stirling numbers of the first kind for specific graph families, providing formulas, recursive identities, and statistical measures like mean and variance to understand cycle distributions.

Contribution

It establishes closed-form expressions, recursive identities, and statistical characterizations for the graphical $r$-Stirling numbers across key graph families, advancing combinatorial and algebraic understanding.

Findings

Explicit formulas for $r$-Stirling numbers on various graphs.

Recursive identities for fundamental graph families.

Mean and variance formulas for cycle distributions.

Abstract

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.