Word posets, with applications to Coxeter groups

We discuss the theory of certain partially ordered sets that capture the structure of commutation classes of words in monoids. As a first application, it follows readily that counting words in commutation classes is #P-complete. We then apply the partially ordered sets to Coxeter groups. Some results are a proof that enumerating the reduced words of elements of Coxeter groups is #P-complete, a recursive formula for computing the number of commutation classes of reduced words, as well as stronger bounds on the maximum number of commutation classes than were previously known. This also allows us to improve the known bounds on the number of primitive sorting networks.

💡 Research Summary

The paper introduces a combinatorial framework called “word posets” to model commutation classes of words in monoids. A word poset is a finite partially ordered set P together with a labeling function s:P→S (where S is the alphabet) satisfying two natural conditions: (a) if two labels are equal or do not commute then the corresponding elements of P must be comparable, and (b) if x<y is a cover (no element strictly between them) then the labels either coincide or do not commute. The authors prove that isomorphism classes of word posets with m elements are in bijection with commutation classes of length‑m words (Theorem 1.1). Moreover, linear extensions of a word poset correspond exactly to the individual words in the associated commutation class (Theorem 1.2).

Counting linear extensions of a poset is a classic #P‑complete problem (Brigh‑twell & Winkler, 1991). By the bijection above, counting the number of words in a given commutation class is therefore #P‑complete (Theorem 1.3). This establishes a strong hardness result for any monoid presented by a finite set of relations that are purely commutation relations.

The second part of the paper applies the word‑poset machinery to Coxeter groups. A Coxeter group is generated by a set S with involution relations a²=1 and braid relations of the form (ab…)ₘ=(ba…)ₘ (including the commutation case m=2). A word is reduced if no shorter word represents the same group element; all reduced words for a fixed element w have the same length ℓ(w). The authors define “reduced word posets” for w, i.e. word posets that encode the commutation classes of reduced words, and denote the set of all such posets by WP(w).

A key observation is that the left descent set D(w)= {a∈S | ℓ(aw)<ℓ(w)} can be computed in polynomial time, and that WP(w) can be built recursively from the families WP(aw) for a∈D(w). Adding a new minimal element labelled a and ordering it below all existing elements whose labels do not commute with a yields a bijection between WP(aw) and a subset of WP(w). This leads to an inclusion–exclusion recurrence for the number of reduced‑word commutation classes C(w)=|WP(w)|:

C(w)=∑_{∅≠T⊂D(w), T independent} (−1)^{|T|+1} C(Tw).

(Theorem 2.2). Here “independent” means that the elements of T pairwise commute, and Tw denotes the product of the letters in T placed on the left of w.

Using this recurrence the authors obtain a substantially stronger upper bound on C(w) than the trivial n^{ℓ(w)} (where n=|S|). Theorem 2.3 shows C(w) ≤ 2^{(3/2)ℓ(w)} for any Coxeter group element with ℓ(w)>0. This improves on the previously known bounds of 3^{ℓ(w)} for the symmetric group and about 2.49^{ℓ(w)} for large lengths.

They also study the extremal function M(k)=max_{|w|=k} C(w), i.e. the maximum possible number of commutation classes among all Coxeter‑group elements of length k. Exact values are given for k≤6, and asymptotic bounds are proved:

½·log 3 ≤ lim_{k→∞} (log M(k))/k ≤ ½·log 3,

suggesting that the limit exists and equals ½·log 3. Theorem 2.4 states that any extremal element can be realized in a finite Coxeter group, so M(k) can be computed by a terminating algorithm.

The final section connects these results to primitive sorting networks. For the longest element w₀ of the finite Coxeter group of type A_{n−1} (the reverse permutation in S_n), the number of reduced‑word commutation classes C(w₀) equals the number P(n) of primitive sorting networks on n wires, which also counts rhombus tilings of a 2n‑gon. Using the recurrence from Theorem 2.2 the authors compute P(12)=2 894 710 651 370 536, extending the previously known sequence (OEIS A006245) beyond n=11. They derive asymptotic inequalities

0.5394… ≤ lim_{n→∞} (log P(n))/k_n ≤ ½·log 3 ≈ 0.5493,

where k_n = n(n−1)/2, and conjecture that the upper bound is tight.

In summary, the paper provides a unified combinatorial model (word posets) that translates problems about commutation classes into counting linear extensions of posets, thereby inheriting known complexity results. It shows that counting reduced words in any Coxeter group is #P‑complete, supplies a recursive inclusion‑exclusion formula for the number of commutation classes, improves exponential upper bounds, and applies these findings to obtain new exact counts and asymptotic estimates for primitive sorting networks. The work bridges combinatorial poset theory, algebraic combinatorics of Coxeter groups, and computational complexity, opening avenues for further exploration of enumeration problems in related algebraic structures.