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 paper establishes a lower bound of n(n-1)/2 for the diameter of the Cayley graph of the symmetric group S_n generated by three specific permutations, advancing understanding of the group's structure.

Contribution

It provides a new lower bound on the diameter of the Cayley graph of S_n with three particular generators, improving previous bounds and insights into the group's complexity.

Findings

Lower bound of n(n-1)/2 for the Cayley graph diameter

Derived from decomposition complexity of specific elements

Enhances understanding of symmetric group structure

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 $(12), (12 \dots n), (1n \dots 2)$.