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

Daniel Yaqubi, Madjid Mirzavaziri

TL;DR

This work introduces graphical $r$-Stirling numbers of the first kind ${G \brack k}_r$, unifying restricted permutation cycle structures with graph-based cycle partitions. It develops explicit closed forms, recurrences, and $r$-cycle polynomials $\mathcal{C}_r(G,x)$ for families such as $P_n$, $C_n$, $S_n$, $W_n$, and $F_n$, and derives mean and variance formulas for the cycle-count distributions, including asymptotic normality in several regimes. The analysis leverages bridge-decomposition, generating polynomials, and Fibonacci/Lucas polynomials, connecting algebraic properties (real-rootedness, log-concavity) with probabilistic limit laws. The results provide a robust framework for restricted cycle decompositions in graphs and pave the way for further exploration of asymptotics, unimodality, and real-rootedness across broader graph classes.

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.

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

TL;DR

This work introduces graphical -Stirling numbers of the first kind , unifying restricted permutation cycle structures with graph-based cycle partitions. It develops explicit closed forms, recurrences, and -cycle polynomials for families such as , , , , and , and derives mean and variance formulas for the cycle-count distributions, including asymptotic normality in several regimes. The analysis leverages bridge-decomposition, generating polynomials, and Fibonacci/Lucas polynomials, connecting algebraic properties (real-rootedness, log-concavity) with probabilistic limit laws. The results provide a robust framework for restricted cycle decompositions in graphs and pave the way for further exploration of asymptotics, unimodality, and real-rootedness across broader graph classes.

Abstract

This paper investigates the \textbf{graphical -Stirling numbers of the first kind}, denoted by , which enumerate partitions of a vertex set into disjoint cycles such that specified vertices occupy distinct blocks. We establish closed-form expressions and recursive identities for fundamental graph families, including \textbf{Path} (), \textbf{Cycle} (), \textbf{Star} (), \textbf{Wheel} (), and \textbf{Fan} () 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 -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.
Paper Structure (11 sections, 51 theorems, 124 equations, 2 figures, 4 tables)

This paper contains 11 sections, 51 theorems, 124 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries and notation
  3. General Statements and Graph Operations
  4. I. The Case $k=1$ (Hamiltonian Cycle):
  5. II. The Case $k \geq 2$.
  6. Graphical $r$-Stirling Numbers of the First Kind
  7. Moments of the Graphical Stirling Distribution
  8. Dense Graph Theorem
  9. Unrestricted case.
  10. Graphical $r$-Cycle Polynomial of the Double Star
  11. Further Work and Open Problems

Key Result

Lemma 3.2

Let $G$ be a simple graph. The graphical Stirling numbers ${G \brack k}$ satisfy the following recursive and structural identities:

Figures (2)

  • Figure 1: The wheel graph $W_4$ with hub $v_0$ and perimeter cycle $C_4$.
  • Figure 2: Visual representation of graph operations at vertex $v_1$.

Theorems & Definitions (113)

  • Definition 2.1
  • Remark 2.2
  • Example 2.3
  • Example 3.1
  • Lemma 3.2
  • proof
  • Corollary 3.3
  • Lemma 3.4
  • proof
  • Lemma 3.5: Kaplansky's Lemma for Circular Arrangements
  • ...and 103 more