A Lower Bound for the Diameter of Cayley Graph of the Symmetric Group $S_n$ Generated by $(12), (12 \dots n), (1n \dots 2)$

TL;DR

This work studies the diameter of the Cayley graph of the symmetric group $S_n$ with the generating set $\{(12), (12\dots n), (1n\dots 2)\}$ under unit edge weights. It develops a reversal-based distance analysis, introducing the elements $s=[n\ n-1\ \dots\ 1]$ and $r=[2\ 3\ \dots\ n\ 1]$, and proves lemmas giving exact lower bounds on the decomposition length of certain reversals, quantified by formulas like $\text{dist}(\pi,\xi r^{\left\lfloor j/2\right\rfloor-1})=j(j-1)-1$. By combining these bounds and using the Lee distance for transitions between powers of $r$, the paper establishes a universal lower bound $\frac{n(n-1)}{2}$ for the diameter. This result tightens our understanding of worst-case permutation decomposition in this generating set and complements existing bounds in the literature on Cayley graph diameters for $S_n$.

Abstract

Let us denote elements of the symmetric group $S_n$ using square brackets for the one-line notation. Cycles will be represented using parentheses, following the standard cycle notation. Under this convention, the full reversal of the identity element $()$ is the element $s = [n\ n-1 \dots 1]$. In the present work, we obtain a lower bound on the decomposition complexity of elements $s(1n \dots 2)^{i}$ into the generators $(12), (12 \dots n), (1n \dots 2)$, where $i$ ranges over the set $\{1,2,\dots,n\}$. As a consequence, we derive the lower bound $n(n-1)/2$ for the diameter of Cayley graph of the group $S_n$ generated by $(12), (12 \dots n), (1n \dots 2)$.

Theorems & Definitions (6)

• Lemma 1
• Proof 1
• Theorem 1
• Proof 2
• Theorem 2
• Proof 3