[Paper Review] On the number of commutation classes of the longest element in the symmetric group

Hugh Denoncourt, Dana C. Ernst+1 more|arXiv (Cornell University)|Jan 25, 2016

Coding theory and cryptography4 citations

TL;DR

This paper investigates the number of commutation classes for the longest element in the symmetric group $S_n$, a longstanding open problem first posed by Knuth in 1992. Using connections to heaps, sorting networks, rhombic tilings, and pseudoline arrangements, the authors establish that the number of commutation classes $c_n$ corresponds to multiple combinatorial objects and provide the first 15 known values of $c_n$, with the best current upper bound of $c_n \leq 2^{0.6571n^2}$ for large $n$. The key contribution is the identification of $c_n$ with several rich combinatorial structures, deepening understanding of reduced word equivalence in Coxeter groups.

ABSTRACT

Using the standard Coxeter presentation for the symmetric group $S_n$, two reduced expressions for the same group element are said to be commutation equivalent if we can obtain one expression from the other by applying a finite sequence of commutations. The resulting equivalence classes of reduced expressions are called commutation classes. How many commutation classes are there for the longest element in $S_n$?

Motivation & Objective

• To determine the number of commutation classes for the longest element in the symmetric group $S_n$, a problem originally posed by Knuth in 1992.
• To establish connections between commutation classes of the longest element and other combinatorial objects such as heaps, sorting networks, and rhombic tilings.
• To provide the first 15 known values of $c_n$, the number of commutation classes for $S_n$, and improve the upper bound on $c_n$.
• To clarify the combinatorial significance of $c_n$ by showing its equivalence to multiple well-known structures in discrete mathematics.

Proposed method

• The authors use the standard Coxeter presentation of $S_n$ to define reduced expressions and equivalence via commutations and braid moves, following Matsumoto’s Theorem.
• They define commutation classes as the equivalence classes of reduced expressions under single commutations, and study their structure via the Coxeter system of type $A_{n-1}$.
• The paper leverages known bijections between commutation classes and combinatorial objects: heaps, primitive sorting networks, rhombic tilings of a $2n$-gon, oriented matroids of rank 3, and pseudoline arrangements.
• The authors compute $c_n$ for $n \leq 15$ using these bijections and existing enumeration techniques, and apply results from Felsner and Valtr to derive the upper bound $c_n \leq 2^{0.6571n^2}$.
• They use lattice point representations of heaps and visualizations of sorting networks and rhombic tilings to illustrate the correspondence and verify small cases like $c_4 = 8$.

Experimental results

Research questions

• RQ1 How many commutation classes exist for the longest element in $S_n$, and what is the growth rate of this number?
• RQ2 What combinatorial structures are in bijection with the commutation classes of the longest element in $S_n$?
• RQ3 Can the number of commutation classes $c_n$ be computed for larger $n$, and what upper bounds can be established?
• RQ4 How do the known values of $c_n$ for $n \leq 15$ inform the asymptotic behavior of $c_n$?

Key findings

• The number of commutation classes $c_n$ for the longest element in $S_n$ is equal to the number of heaps for $w_0$, as established by a bijection in [13, Proposition 2.2].
• For $n=4$, there are exactly 8 commutation classes, confirmed by enumerating 16 reduced expressions and grouping them under commutation equivalence.
• The first 15 values of $c_n$ are now known: $1, 1, 2, 8, 62, 908, 24698, 1232944, 112018190, 18410581880$, with $c_{15}$ being the largest computed so far.
• The best known upper bound for $c_n$ is $c_n \leq 2^{0.6571n^2}$ for sufficiently large $n$, derived from results on pseudoline arrangements by Felsner and Valtr.
• Commutation classes of $w_0$ are in bijection with primitive sorting networks on $n$ elements, as well as with rhombic tilings of a regular $2n$-gon with unit rhombi.
• The problem remains open for other finite Coxeter groups, with little known about $c_n$ beyond the symmetric group $S_n$.