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

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

• The paper establishes that the Cayley graph diameter of Sₙ is bounded below by n(n-1)/2 using specified generators.
• It employs inversion counting and reversal decomposition techniques to derive the minimal word length complexity.
• The result sharpens prior conjectures and has significant implications for permutation sorting and group-theoretic algorithms.

Formal Summary of “A Lower Bound for the Diameter of Cayley Graph of the Symmetric Group $S_n$ Generated by $(12), (12 \dots n), (1n \dots 2)$” (2601.08715)

Problem Statement and Context

The paper investigates combinatorial and metric properties of the Cayley graph $\Gamma$ of the symmetric group $S_n$ under the generating set $\{(12), (12\dots n), (1n \dots 2)\}$. The diameter of this Cayley graph corresponds to the maximal minimal word length for expressing an arbitrary permutation via these generators. This metric is closely related to worst-case decomposition complexity and has algorithmic implications for sorting and group-theoretic computations.

Definitions and Preliminaries

The symmetric group $S_n$ is considered with two notations for permutations: the compact one-line notation $[\sigma(1)\, ...\, \sigma(n)]$ and the classical cycle notation. The generating set includes a transposition $(12)$, a full cycle $(12\dots n)$, and its reverse $(1n\dots 2)$.

The Cayley graph is edge-weighted uniformly, and the word metric $\textup{dist}(\cdot,\cdot)$ acts as the shortest path length between permutations, with the graph diameter defined as the maximum such distance among all pairs. The analysis leverages the correspondence between reversals of permutation substrings and certain group elements, utilizing orbits under cyclic shifts and the induced Lee metric.

Main Results

Lemma: Minimal Decomposition Complexity for Subsequence Reversal

A key combinatorial lemma establishes that for $2 \leq j \leq n$, transforming a permutation $\pi$ into its $(\pi_1,\pi_j)$-reversal requires at least $j(j-1)-1$ steps in the Cayley graph. This is based on the counting of inversions induced by reversals and the optimality of specific generator sequences for performing swaps with minimal steps.

Theorem: Lower Bound for Word-Length Complexity

Building on the lemma, the author proves that for elements of the form $s(1n\dots2)^i$, where $s = [n\,n-1\,\dots\,1]$ is the full reversal and $i \in \{1,2,\dots,n\}$, the minimal decomposition length into the prescribed generators reaches $n(n-1)-1$ twice, plus a function in $n$ and $i$. This covers permutations expected to induce the extremal diameter, in particular $s(1n\dots2)^2$.

Final Lower Bound for Diameter

Aggregating the preceding results, the paper derives an explicit lower bound for the diameter of the Cayley graph $\Gamma$:

$\textup{diam}(\Gamma) \geq \frac{n(n-1)}{2}$

This confirms and sharpens prior conjectures, specifically validating that certain shifted reversals attain the lower bound for decomposition complexity.

Relation to Prior Work

The results complement earlier studies targeting Cayley graph diameters under various generator sets and edge weightings. For example, when restricting generator weights (e.g., making the cycle generator “cost” infinity or zero), the diameter changes dramatically, as noted in (Adin et al., 20 Feb 2025) and (Chervov et al., 25 Feb 2025). The present paper fixes all weights to unity and achieves tighter lower bounds compared to those previously established in the literature.

Implications and Speculative Outlook

The explicit lower bound $\frac{n(n-1)}{2}$ for Cayley graph diameter with three fundamental generators provides benchmark complexity metrics for permutation group decompositions. It establishes a sharp estimate for worst-case word lengths and has consequences for the efficiency of permutation sorting algorithms and related path-finding routines in group-theoretic reinforcement learning frameworks (cf. (Chervov et al., 25 Feb 2025)). From a theoretical perspective, the techniques elucidated—including fine-grained reversal decompositions and inversion counting arguments—may generalize to other classes of finite groups and alternative generator systems, opening avenues for further research in algebraic combinatorics and computational group theory.

Speculatively, these diameter bounds may inform the design of efficient canonical representations for permutations in memory and impact cryptographic protocols relying on permutation group structures. Further investigations could pursue asymptotic refinements, explore connections to Gray code constructions, or extend the approach to weighted Cayley graphs and other algebraic metrics.

Conclusion

The paper rigorously establishes that for $S_n$ generated by $(12)$, $(12 \dots n)$, and $(1n \dots 2)$, the Cayley graph diameter is bounded below by $\frac{n(n-1)}{2}$. This result is achieved via detailed combinatorial breakdowns of permutation reversals and optimal generator sequences, representing a precise measure of decomposition complexity in this foundational group-theoretical context.