On the Number of Commutation Classes of the Longest Element in the Symmetric Group

Using the standard Coxeter presentation for the symmetric group Sn, 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 Sn? Original proposer of the open problem: Donald E. Knuth The year when the open problem was proposed : 1992 [11, §9] A Coxeter system is a pair (W,S) consisting of a distinguished (finite) set S of generating involutions and a group W = 〈S | (st) = e for m(s, t) < ∞〉, called a Coxeter group, where e is the identity, m(s, t) = 1 if and only if s = t, and m(s, t) = m(t, s). It turns out that the elements of S are distinct as group elements and that m(s, t) is the order of st. Since the elements of S have order two, the relation (st) = e can be written to allow the replacement sts · · · } {{ } m(s,t) 7→ tst · · · } {{ } m(s,t) which is called a commutation if m(s, t) = 2 and a braid move if m(s, t) ≥ 3. Given a Coxeter system (W,S), a word w = sx1sx2 · · · sxm in the free monoid S ∗ is called an expression for w ∈ W if it is equal to w when considered as a group element. If m is minimal among all expressions for w, the corresponding word is called a reduced expression for w. In this case, we define the length of w to be l(w) = m. According to [8], every finite Coxeter group contains a unique element of maximal length, which we refer to as the longest element and denote by w0. Let (W,S) be a Coxeter system and let w ∈ W . Then w may have several different reduced expressions that represent it. However, Matsumoto’s Theorem [7, Theorem 1.2.2] 2010 Mathematics Subject Classification. 05. 1 2 DENONCOURT, ERNST, AND STORY 321323 323123 312312 132312 312132 132132 321232 232123 123121 121321 231231 213231 231213 213213 123212 212321 Figure 1. Reduced expressions and the corresponding commutation classes for the longest element in S4. states that every reduced expression for w can be obtained from any other by applying a finite sequence of commutations and braid moves. Following [13], we define a relation ∼ on the set of reduced expressions for w. Let w and w be two reduced expressions for w and define w ∼ w if we can obtain w from w by applying a single commutation. Now, define the equivalence relation ≈ by taking the reflexive transitive closure of ∼. Each equivalence class under ≈ is called a commutation class. The Coxeter system of type An−1 is generated by S(An−1) = {s1, s2, . . . , sn−1} and has defining relations (i) sisi = e for all i; (ii) sisj = sjsi when |i−j| > 1; and (iii) sisjsi = sjsisj when |i−j| = 1. The corresponding Coxeter group W (An−1) is isomorphic to the symmetric group Sn under the correspondence si 7→ (i, i+1). It is well known that the longest element in Sn is given in 1-line notation by w0 = [n, n− 1, . . . , 2, 1] and that l(w0) = ( n 2 ) . Let cn denote the number of commutation classes of the longest element in Sn. The longest element w0 in S4 has length 6 and is given by the permutation (1, 4)(2, 3). There are 16 distinct reduced expressions for w0 while c4 = 8. The 8 commutation classes for w0 are given in Figure 1, where we have listed the reduced expressions that each class contains. Note that for brevity, we have written i in place of si. In [12], Stanley provides a formula for the number of reduced expressions of the longest element w0 in Sn. However, the following question is currently unanswered. Open Problem. What is the number of commutation classes of the longest element in Sn? To our knowledge, this problem was first introduced in 1992 by Knuth in Section 9 of [11], but not using our current terminology. A more general version of the problem appears in Section 5.2 of [9]. In the paragraph following the proof of Proposition 4.4 of [14], Tenner explicitly states the open problem in terms of commutation classes. According to sequence A006245 of The On-Line Encyclopedia of Integer Sequences [1], the first 10 values for cn are 1, 1, 2, 8, 62, 908, 24698, 1232944, 112018190, 18410581880. To COMMUTATION CLASSES OF THE LONGEST ELEMENT IN THE SYMMETRIC GROUP 3 1 2 2 3 3 3 1 1 2 2 3 3 1 2 2 2 3 3 1 2 2 2 3 3 3 2 2 1 1 1 1 1 2 2 3 3 1 1 2 2 2 3 3 2 2 2 1 1 Figure 2. Heaps for the longest element in S4. Figure 3. Minimal ladder lotteries corresponding to the primitive sorting networks on 4 elements. date, only the first 15 terms are known. The current best upper-bound for cn was obtained by Felsner and Valtr. They prove that for sufficiently large n, cn ≤ 2 0.6571n [5, Theorem 2], although their result is stated in terms of arrangements of pseudolines. The commutation classes of the longest element of the symmetric group are in bijection with a number of interesting objects. It turns out that cn is equal to the number of • heaps for the longest element in Sn [13, Proposition 2.2]; • primitive sorting networks on n elements [2, 10, 11, 15, 16]; • rhombic tilings of a regular 2n-gon (where all side lengths of the rhombi and the 2n-gon are the same) [3, 14]; • oriented matroids of rank 3 on n elements [6, 9]; • arrangements of n pseudolines [4, 5, 11]. In Figure 2, we have drawn lattice point representations of the 8 heaps that correspond to the commutation classes for the longest element in S4. Note that our heaps are sideways versions of the heaps that usually appear in the literature. The minimum ladder lotteries (or ghost legs) corresponding to the 8 primitive sorting networks on 4 elements are provided in Figure 3. The 8 distinct rhombic tilings of a regular octagon are depicted in Figure 4. 4 DENONCOURT, ERNST, AND STORY Figure 4. Rhombic tilings of a regular octagon. Very little is known about the number of commutation classes of the longest element in other finite Coxeter groups.