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Maximum number of one-element commutation classes of a permutation

Ricardo Mamede, José Luis Santos, Diogo Soares

TL;DR

The paper proves a universal bound $|R_{ullet}(\sigma)|\le 4$ for all permutations $\sigma\in\mathfrak{S}_{n+1}$, showing that one-element commutation classes are highly constrained. It achieves this via a careful structural analysis that partitions words into oscillations and non-oscillations, employing line diagrams, endpoints, and Tenner's criteria, with an inductive argument on permutation length. As a consequence, the authors establish $|C(\sigma)|\le \tfrac{1}{2}|R(\sigma)|+1$, providing a tight link between the total number of reduced words and their commutation classes (and nearly resolving Elder's conjecture). The results reveal a surprising rigidity in one-element commutation classes, and the methods offer a framework for exploring analogous questions in other Coxeter types and for identifying permutations with such classes. Together, these contributions advance understanding of the combinatorics of reduced words and their commutation structures in symmetric groups.

Abstract

In this paper, we provide an upper bound for the number of one-element commutation classes of a permutation, that is, the number of reduced words in which no commutation can be applied. Using this upper bound, we prove a conjecture that relates the number of reduced words with the number of commutation classes of a permutation.

Maximum number of one-element commutation classes of a permutation

TL;DR

The paper proves a universal bound for all permutations , showing that one-element commutation classes are highly constrained. It achieves this via a careful structural analysis that partitions words into oscillations and non-oscillations, employing line diagrams, endpoints, and Tenner's criteria, with an inductive argument on permutation length. As a consequence, the authors establish , providing a tight link between the total number of reduced words and their commutation classes (and nearly resolving Elder's conjecture). The results reveal a surprising rigidity in one-element commutation classes, and the methods offer a framework for exploring analogous questions in other Coxeter types and for identifying permutations with such classes. Together, these contributions advance understanding of the combinatorics of reduced words and their commutation structures in symmetric groups.

Abstract

In this paper, we provide an upper bound for the number of one-element commutation classes of a permutation, that is, the number of reduced words in which no commutation can be applied. Using this upper bound, we prove a conjecture that relates the number of reduced words with the number of commutation classes of a permutation.
Paper Structure (8 sections, 18 theorems, 26 equations, 9 figures, 1 table)

This paper contains 8 sections, 18 theorems, 26 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. An upper bound for $|R_{\bullet}(\sigma)|$
  4. Oscillations
  5. The non-oscillation case and the proof of Theorem \ref{['thm1']}
  6. An application
  7. Open problems and further directions
  8. Acknowledgments

Key Result

Theorem A

Let $n\geq 1$. Then, for all $\sigma\in \mathfrak{S}_{n+1}$ we have $|R_{\bullet}(\sigma)|\leq 4$.

Figures (9)

  • Figure 1: Line diagram of the word $2345432123456765434= {$$} {$x$} {$2$} \hbox{[-1]{}} {2}\,\widehat{5}\, {$$} {$x$} {$1$} \hbox{[-1]{}} {1}\,\widehat{7}\, {$$} {$x$} {$3$} \hbox{[-1]{}} {3}\,\widehat{4}$.
  • Figure 2: Line diagram of the word ${$$} {$x$} {$2$} \hbox{[-1]{}} {2}\,\widehat{5}\, {$$} {$x$} {$1$} \hbox{[-1]{}} {1}\,\widehat{7}\, {$$} {$x$} {$3$} \hbox{[-1]{}} {3}\,\widehat{6}$ with a factor with repeated segments highlighted in red.
  • Figure 3: Possible line diagrams of a word containing a factor with repeated segments. On the left (resp. right) line diagram, $x$ satisfies $i\leq x<j\leq y$ (resp. $y\leq i<x\leq j$).
  • Figure 4: Line diagram of the word $\widehat{3} {$$} {$x$} {$1$} \hbox{[-1]{}} {1}\widehat{4} {$$} {$x$} {$1$} \hbox{[-1]{}} {1}\widehat{2}$ with a factor with symmetric segments highlighted in red.
  • Figure 5: Line diagrams for the one-element commutation classes of $\sigma$.
  • ...and 4 more figures

Theorems & Definitions (41)

  • Theorem A: Theorem \ref{['bound']}
  • Theorem B: Theorem \ref{['conjecture']}
  • Definition 2.1: OECTenner
  • Definition 2.2: OEC
  • Definition 2.3: Tenner
  • Theorem B: Tenner
  • Definition 2.4: OEC
  • Definition 2.5: OEC
  • Theorem B: OEC
  • Lemma B: TennerRep
  • ...and 31 more