On random shifted standard Young tableaux and 132-avoiding sorting networks

We study shifted standard Young tableaux (SYT). The limiting surface of uniformly random shifted SYT of staircase shape is determined, with the integers in the SYT as heights. This implies via properties of the Edelman-Greene bijection results about random 132-avoiding sorting networks, including limit shapes for trajectories and intermediate permutations. Moreover, the expected number of adjacencies in SYT is considered. It is shown that on average each row and each column of a shifted SYT of staircase shape contains precisely one adjacency.


Introduction
A shifted standard Young tableau (SYT) of staircase shape is an increasing filling of the shifted diagram of the partition (n − 1, . . . , 2, 1) with the integers 1, 2, . . . , n 2 . See Figure 1a for an example and Section 2 for the exact definition. Shifted diagrams and tableaux are important combinatorial objects that appear in various contexts. In representation theory shifted Young diagrams correspond to projective characters of the symmetric group, and shifted tableaux lend themselves to being studied via RSK-type methods [32,40]. In the theory of partially ordered sets, shifted diagrams alongside non-shifted Young diagrams and rooted trees form the three most interesting families of d-complete posets, which are in turn connected to fully commutative elements of Coxeter groups [41,28]. The most salient property of d-complete posets is the fact that their linear extensions (in our case shifted SYT) are enumerated by elegant product formulas involving hook lengths, see [28,23]. Shifted diagrams also appear as order filters in the root poset of type B n , and shifted SYT play an important role in the enumeration of reduced words of elements of the Coxeter group of type B n [15,18]. Moreover, as is topical in this paper, shifted SYT are also relevant to the study of certain reduced words in the symmetric group. Recently, Elizalde and Roichman related shifted diagrams and tableaux to unimodal permutations [7].
The topics of this paper can be divided into three parts. In Section 3 we study the surface obtained by viewing the integers in random SYT as heights. The study of limit phenomena for partitions and tableaux is an active field of research combining methods from combinatorics, probability theory and analysis. We refer to [30] for a general survey. Shifted objects have been treated as well, for example Ivanov [16]  (a) A shifted SYT of staircase shape.
(b) The limit shape of uniformly random shifted SYT of staircase shape.

Figure 1
measure on shifted diagrams. In the present paper we determine the limiting surface for uniformly random shifted SYT of staircase shape, see Figure 1b and Theorem 3.7. The deduction of our results relies on a paper by Pittel and Romik [27] where the limit shape for random rectangular SYT is determined. In fact, we end up with the same variational problem, and the limit surface for shifted staircase SYT is half of the surface for square SYT. This analogy is in part explained by a combinatorial identity in Proposition 3.1 relating shifted and non-shifted tableaux. There are few shapes for which the limit surface has been explicitly described previously. As far as we know the only other case is that of staircase SYT, where again the same limit surface appears, but cut along a different diagonal [1]. However, it was pointed out to us by one of the referees that Biane [2] provides a general method for computing such limits for SYT. Recently, Matsumoto and Śniady developed an analogous framework for shifted SYT which, combined with the results of [2] and [27], allowed them to partially recover Theorem 3.7 in [22,Sec. 8.4]. Results of this type have applications in other fields of mathematics such as geometric complexity theory [26]. Secondly, we study 132-avoiding sorting networks, which are by definition reduced words w 1 . . . w ( n 2 ) of the reverse permutation such that s w1 · · · s w k is 132-avoiding for any 1 k n 2 . These objects have received considerable recent interest and also appear in different guises, for example as chains of maximum length in the Tamari lattice [3]. Fishel and Nelson [9] showed that 132-avoiding sorting networks are in bijection with shifted SYT of staircase shape via the Edelman-Greene correspondence. This has been rediscovered several times [36,5,20]. They are also in bijection with reduced words of the signed permutation (−(n − 1), −(n − 2), . . . , −1) via the shift s i → s i−1 as was remarked in [36,Sec. 1.3]. In Sections 4 and 5 the Edelman-Greene bijection is used to transfer the limit shape of shifted SYT to determine the limit shapes of intermediate permutations (Theorem 4.2) and trajectories (Theorem 5.4) in random 132-avoiding sorting networks. These results are motivated by a remarkable paper of Angel, Holroyd, Romik and Virág [1] that contains a number of tantalising conjectures about random sorting networks, now proven by Dauvergne [4]. See Section 2 for a description of some of them. Our results are a parallel to their (former) conjectures restricted to a subclass of random sorting networks. The 132-avoiding sorting networks are much easier to treat than the general case. The main reason for Algebraic Combinatorics, Vol. 3 #6 (2020) On random shifted SYT and 132-avoiding sorting networks this is that the Edelman-Greene bijection simplifies significantly when it is restricted to 132-avoiding sorting networks.
We remark that the limit surface for shifted SYT of staircase shape contains complete information on the limit surface for SYT of square shape. This suggests the perhaps less intuitive idea that the relatively small subset of 132-avoiding sorting networks contains a lot of information on random sorting networks in general.
The third set of results is obtained in Section 6 and concerns patterns in 132avoiding sorting networks. We first observe that adjacencies in a shifted SYT (that is, integers i and i + 1 in neighbouring cells) translate directly to adjacencies in a 132avoiding sorting network (that is, j and j + 1 next to each other in the reduced word). Corollary 6.5 asserts that the expected number of adjacencies in each column and each row in a shifted SYT of staircase shape is exactly 1. The proof uses promotion and evacuation techniques very similar to the methods used by Schilling, Thiéry, White and Williams [36] to derive results on Yang-Baxter moves (that is, patterns of the form j(j ± 1)j) in 132-avoiding sorting networks. Related results on general sorting networks are due to Reiner [29] and Tenner [43].

Background
In this section we fix notation and review some facts about partitions, tableaux and random sorting networks.

Partitions and tableaux.
A partition is a weakly decreasing sequence λ = (λ 1 , . . . , λ n ) of positive integers. If a partition is strictly decreasing it is called strict. The sum λ i is called the size of the partition λ and is denoted by |λ|. The number of entries λ i is called the length of the partition and is denoted by (λ). Define the staircase partition as ∆ n = (n − 1, . . . , 2, 1).
The Young diagram of a partition λ is defined as the set The elements (i, j) are indexed with matrix notation and typically referred to as cells of λ. The conjugate partition λ of the partition λ is the partition corresponding to the Young diagram {(j, i) : (i, j) ∈ λ dg }. Given a strict partition λ we also define its shifted Young diagram as Thus the shifted Young diagram is obtained from the normal Young diagram by shifting rows to the right. These definitions are illustrated in Figure 2. Let u = (i, j) be a cell in the Young diagram of a partition λ. The hook-length of u, denoted by h λ (u), is defined as the number of cells in the same row as u that lie to the right of u plus the number of cells in the same column as u that lie weakly below u (that is, including u itself). Thus, h λ (u) = λ i − j + λ j − i + 1. Given a cell u = (i, j) in the shifted Young diagram of a strict partition, we define its shifted hook-length h sh λ (u) as the number of cells in the same row as u that lie to the right of u, or in the same column and weakly below u, plus the number of cells in the (i + j)-th row of the shifted Young diagram. See   Figure 3. The (shifted) Young diagram of λ = (6, 4, 2, 1) with the (shifted) hook-lengths filled into each cell.
Given a cell u = (i, j) ∈ Z 2 , define the north, east, south, and west neighbour of u as is a bijection and T (u) < T (eu) and T (u) < T (su) whenever the respective cells lie in λ dg . Similarly a shifted standard Young tableau of shape λ sh is a bijection T : λ sh → [n] such that T (u) < T (eu) and T (u) < T (su) whenever the respective cells lie in the shifted Young diagram λ sh . Let T n = {T : T is a shifted SYT of shape ∆ sh n }. For example, Figure 1a shows a shifted standard Young diagram of shape ∆ sh 5 . The number of SYT is given be the following nice product formula.
Theorem 2.1 (Hook-length formula [11,44]). Let λ be a partition. Then the number of SYT of shape λ dg is given by .
If λ is strict then the number of shifted SYT of shape λ sh is given by .
The literature offers a variety of proofs of different flavours for Theorem 2.1, for example using hook-walks [13,31] or by means of jeu de taquin [24,8].

2.2.
Random 132-avoiding sorting networks. The reader is referred to [17] for background on pattern avoidance in permutations.
For i ∈ [n − 1] let s i = (i, i + 1) denote the i-th adjacent transposition. The reverse permutation w 0 ∈ S n is defined by w 0 (i) = n − i + 1 for i ∈ [n]. A reduced word of w 0 is a word w = w 1 · · · w N in the alphabet [n − 1] such that w 0 = s w1 · · · s w N .
We will adopt the convention that multiplying a permutation by s i from the right interchanges the numbers in positions i and i + 1 (or swaps columns i and i + 1 of the permutation matrix, see below). This means equivalently that multiplying by s i from the left interchanges the positions of the values i and i + 1 (or swaps rows i and i + 1 of the permutation matrix). As an example consider the word w = 1213, Algebraic Combinatorics, Vol. 3 #6 (2020) corresponding to composing s 1 s 2 s 1 s 3 , which yields the permutation 3241. In terms of permutation matrices, we have where we can see that multiplying by s i from the right corresponds to swapping the columns i and i + 1. Angel, Holroyd, Romik and Virág introduced n-element random sorting networks in [1] as the set of reduced words of the reverse permutation w 0 ∈ S n equipped with the uniform probability measure. In the same paper, Angel et al. pose several striking conjectures about random sorting networks, which have now been proven by Dauvergne [4]. In other words, their 1s occur inside an elliptic region of the matrix. In particular, at half-time the permutation matrix is supported on a disc. Figure 4 provides an illustration.
Every permutation σ ∈ S n lies on the sphere Random sorting networks correspond to paths on the permutahedron. The strongest theorem, [4,Thm. 4] (previously [1, Conj. 3]), which implies both of the previous ones, states that these paths are close to great circles in S n−2 .
This paper considers similar questions restricted to 132-avoiding sorting networks, that is, those reduced words w 1 . . . w N of the reverse permutation in S n such that s w1 · · · s w k is 132-avoiding for all k ∈ [N ]. With a random 132-avoiding sorting network we will refer to uniform distribution on all 132-avoiding sorting networks of a fixed length. Understanding them is easier than random sorting networks since the Edelman-Greene correspondence restricted to 132-avoiding sorting networks has a much simpler description than the general bijection. This is discussed next.
The connection between 132-avoiding sorting networks and shifted SYT is the following. Let w = w 1 . . . w N be a 132-avoiding sorting network. Then, for k ∈ [N ], define a SYT Q w1...w k by letting its j-th column consist of the indices m ∈ [k] such   that w m = j. Furthermore, define a shifted SYT Q → w1...w k by shifting the rows of Q w1...w k . Figure 6 shows an example. . For all n ∈ N, the map w → Q → w is a bijection from n-element 132-avoiding sorting networks to shifted SYT of shape ∆ sh n . The map w → Q w agrees with the restriction of the Edelman-Greene correspondence to 132-avoiding sorting networks.
The same bijection was also described in [20], in [36,Fig. 4] in terms of heaps, and in [5,Prop. 5.2] in terms of descent sets.

The limit shape
In this section we derive a limit shape for random shifted SYT of staircase shape. We may interpret a shifted SYT T ∈ T n as the graph of a function by viewing the entries as heights Our main result, Theorem 3.7, states that by this choice of scaling, the functions L T (n) converge with probability 1 to the surface depicted in Figure 1b, where each T (n) ∈ T n is chosen uniformly at random. The proof of Theorem 3.7 relies heavily on the work of Pittel and Romik [27]. Recall our conventions from Section 2.1. Given a strict partition λ define a partition Λ by letting its Young diagram equal It is easy to see that this really is the Young diagram of a partition. See Figure 7. We call Λ the double of λ. The diagram of Λ is at times, for example in [21], denoted by D(λ). The motivation for this definition is the fact that shifted hook-lengths of the cells in λ sh correspond to hook-lengths of the corresponding cells in Λ dg .
The following proposition is not new and can be obtained by extracting coefficients from an identity of symmetric functions found in [21, Chap. III, Sec. 8, Example 9.(b)]. We include a proof nevertheless since it is short and needs no additional terminology.
Algebraic Combinatorics, Vol. 3 #6 (2020) Proof. We show that This is obvious if λ is a staircase and Λ is a rectangle. In this case , as in Figure 8a, and Figure 8b.   The identity in (2) follows inductively as it is easy to verify that (3) and (4) are preserved when a cell (i, k) with (λ) < k is added to λ sh . Adding a cell to row i increases h sh λ (i, (λ)) by 1 and h Λ (i, i) by 2, hence the equality in (3) still holds. Moreover, since (k, i) and (i, k + 1) are added to Λ dg , the equality in (4) is preserved. For the remaining cells we have Figure 9. The claim then follows from the hook-length formula (Theorem 2.1). The following two results provide an estimate for the probability that a fixed subdiagram λ sh of size k of the shifted staircase contains precisely the entries 1, . . . , k in a shifted SYT of shape ∆ sh n chosen uniformly at random. Let m, n ∈ N, = (n m ), and λ be a partition. If λ dg ⊆ dg define a partition λ by setting This is a 180 • rotation of the skew diagram of λ. Moreover, if λ is strict and λ sh ⊆ ∆ sh n , define a strict partition ∆ n λ by setting which is the reflection of ∆ sh n λ sh through the diagonal i = j. See Figure 10. Lemma 3.2. Let n ∈ N, λ be a strict partition of size k with λ 1 < n, and let P n denote the uniform probability measure on T n . Then where Λ and n are the doubles corresponding to λ and ∆ n respectively.
Proof. The left hand side of (5) is equal to It is not difficult to see that n Λ is the double of ∆ n λ. See Figure 10. Thus (6) can be computed by means of Proposition 3.1 above.
The right hand side of (5) is similar to [27, Eq. (7)]. This allows us to prove an analogue of [27, Lem. 1] for the shifted staircase. Note that n denotes an n × n square in [27], whereas here it is an (n − 1) × n rectangle.
We now describe scaled and rotated boundary functions of partitions. We use the same conventions as Pittel and Romik. Fix n and let λ be a partition with |λ| = k. Define a function γ λ : where by convention λ i = 0 for i > (λ). See Figure 11. It is often more convenient to work with rotated coordinates Define g λ : R → R 0 by Figure 11. The functions γ λ and g λ for the partition (6, 4, 2, 1).
where x(u, v) and y(u, v) are given by the inverse of the transformation in Equation (7). The function g λ is just a rotated version of the function γ λ , however, it has the advantage of being 1-Lipschitz while γ λ is only non-increasing. Note that where implicitly x = x(u, v) and y = y(u, v). The function G λ describes the boundary of the scaled shifted Young diagram λ sh . See Figure 12.
be a sequence such that k/N → α as n → ∞, and let P n denote the uniform probability measure on T n . Then, as n → ∞, uniformly over all strict partitions λ of k with λ 1 < n, where and Λ denotes the double of λ.
Proof. We use Lemma 3.2 and proceed exactly as in the proof of [27,Lem. 1]. Compared to [27] our partitions lie inside an (n − 1) × n rectangle rather than a square, however, the obtained error is small as n tends to infinity. Also note that as n → ∞ by Stirling's approximation, thus these terms do not contribute to the analysis.
Results of the type of Lemma 3.3 lead to a so-called large deviation principle. Suppose that λ is the strict partition of size k with λ 1 < n such that its double Λ minimises the integral I(γ Λ ). If µ is a different strict partition of size k with µ 1 < n then by Lemma 3.3 the probability that µ contains the numbers 1, . . . , k in a random shifted SYT T ∈ T n decays exponentially as µ deviates from λ. This means that the shape formed by the entries 1, . . . , k in a random shifted SYT will be close to the minimising partition λ with high probability. One is therefore lead to the variational problem of identifying the function γ within a certain search space of functions depending on α that minimises the integral I(γ).
A function g : is called α-admissible if it is 1-Lipschitz and satisfies As is explained in [27, Sec. 2.2] our problem is equivalent to the following formulation: For each α ∈ (0, 1) find the unique α-admissible function g which is symmetric, that is, g(−u) = g(u), and minimises the integral The only difference between our situation and the situation in [27] is the fact that our search space of admissible functions g is smaller since we require that Λ is the double of a strict partition. In [27, Sec. 2 and 3] Pittel and Romik show that the variational problem (8) without the assumption g(−u) = g(u) has the unique solutiong α given by is defined as The family of functions g α is illustrated in Figure 12. Since this solution already exhibits the additional symmetryg α (−u) =g α (u) as tan −1 is odd, we may apply it to the shifted case as well.
where (x, y) and (u, v) are related as in (7).
The following lemma collects analytic results on the integral K(g) and the functions g α that are found in [27].
There exists a function c : (2, 3) → R >0 such that for all r ∈ (2, 3) and all α-admissible functions g we have Let (x, y) ∈ (0, 1) × (0, 1), let (u, v) be given as in (7), set α = L(u, v), and set σ(x, y) = min xy, (1 − x)(1 − y) . Then there exist constants c 1 > 0 and c 2 > 0 such that for all β ∈ (0, 1) and all δ < c 2 σ(x, y) 2 we have The last ingredient that is needed to obtain the limit shape for random shifted SYT of staircase shape is a bound on the expected number of entries less than k in the first row of such a tableau. We start with an auxiliary result. Given a partition λ let λ + denote the partition obtained from λ by adding a cell to the first row, that is, λ + = (λ 1 + 1, λ 2 , . . . , λ (λ) ).
Lemma 3.5. Let m, n ∈ N, = (n m ) be a rectangle and λ a partition such that Proof. This was proved by Pittel and Romik in the case where is an n times n square [27,Eq. (71)]. The proof of the generalisation to rectangles relies on the same idea.
After even more cancellations (10) is equal to which we rewrite as where A denotes the set of cells u ∈ λ dg such that eu, su / ∈ λ dg , B denotes the set of cells u ∈ dg λ dg such that nu, wu / ∈ ( dg λ dg ), and which can be written as where A and B are defined as above, and otherwise.
The claim now follows from the observation that almost all factors that appear in (11) and (12) cancel.
The following lemma provides us with an analogue of [27, Eq. (75)] for the shifted case. Given n, k ∈ N with k < N , let I n,k : T n → {0, 1} denote the random variable that takes the value 1 if the entry k is contained in the first row, and 0 otherwise. Moreover, let J n,k = k i=1 I n,i denote the number of entries at most k in the first row of a shifted SYT.
Lemma 3.6. Let k, n ∈ N with 1 < k < N , and let E n denote the expected value with respect to the uniform probability measure P n on T n . Then Algebraic Combinatorics, Vol. 3 #6 (2020) Proof. Our proof is very similar to the first part of the proof of [27,Lem. 10]. Let Y n,k denote the set of all strict partitions λ of size k with λ 1 < n − 1. Note that Using the fact that µ → f sh µ · f sh ∆n µ /f sh ∆n defines a probability measure on the set of strict partitions µ of size k with µ 1 < n, and the convexity of the square function we obtain Let L, M and n denote the doubles of λ, λ + and ∆ n , respectively. Proposition 3.1 yields (14) f Using M = ((L + ) ) + and Lemma 3.5 twice we obtain .
Inserting this into (14) we obtain The claim now follows from To see this note that the first factor in the summand is the probability that the entries 1, . . . , k − 1 of a random shifted SYT of shape ∆ sh n are contained precisely in the shifted diagram λ sh . Hence the value λ 1 in the summand counts the number of entries less than k that appear in the first row.
We now prove the limit shape theorem for shifted SYT of staircase shape chosen uniformly at random. Our result is an analogue of [27, Thm. 1]. The obtained limit shape is the same as the limit shape for random SYT of square shape except that the domain is restricted from a square to a triangle.
In particular (16) provides point-wise convergence to the limit surface, while (17) specifies the rate of convergence if we assume a sufficient distance to the short sides of the triangle.

On random shifted SYT and 132-avoiding sorting networks
Theorem 3.7. For n ∈ N let ∆ n denote the staircase partition of size N = n 2 , T n the set of shifted SYT of shape ∆ sh n , and P n the uniform probability measure on T n . Then for all > 0 (16) lim n→∞ P n T ∈ T n : max Moreover for all p ∈ (0, 1/2) and all q ∈ (0, p/2) such that 2p + q < 1 (17) lim n→∞ P n T ∈ T n : max Proof. The first part (16) (17), for which [27] seems to contain a small inaccuracy. Let p (k) denote the number of strict partitions of size k. Then Fig. I.9].
Given k = αN and a tableau T ∈ T n let λ T,k denote the partition with shifted Young diagram λ sh T,k = T −1 ([k]). Let Λ T,k denote the double of λ T,k . Note that given the double Λ of a strict partition λ of k, the function g Λ is 1-Lipschitz but not αadmissible, since However, we can always choose an α-admissible functionĝ Λ such that This is needed below for using Lemma 3.4 (ii). There exists a constant C > 0 such that for all r ∈ (2, 3) and all 1 > 0 as n → ∞. For (i, j) ∈ ∆ sh n set β = L(i/n, j/n). Note that if (u, v) corresponds to (x, y) = (i/n, j/n), theng β (u) = v, and since T (λ T,T (i,j) ) = [T (i, j)], G λ T ,T (i,j) (u) = v by the Algebraic Combinatorics, Vol. 3 #6 (2020) definition of G. Given T ∈ T n set α T = T (i, j)/N . For all r ∈ (2, 3) and all δ > 0 that satisfy δ < c 2 σ(i/n, j/n) 2 , if n is large enough, then (20) exp Cn − c(r) 2 Suppose δ = n −p and σ(i/n, j/n) > n −q for some p > 0 and q > 0 such that p > 2q and Then for n large enough δ = n −p < n −2q < c 2 σ(i/n, j/n) 2 and δc 1 2 σ(i/n, j/n) > c 1 2 For all 2 > 0 there exists 3 > 0 such that the choice

Thus since the number of cells in ∆ sh
n is only quadratic in n, we obtain (17) by taking the union bound in (21) over all possible cells.  Recall the definition of the extensiong α of g α to the full interval [− √ 2/2, √ 2/2] in (9). See also Figure 13a. The theorem below uses Theorem 4.1 to describe the limit shape of intermediate permutation matrices in terms of the extended level curvesg α . Since we use diagrams of permutations, we have to do this in two parts -first for the indices j ∈ [n] for which σ αN (j) σ αN (1), that is, the columns of M (σ αN ) in which the 1-entry is weakly above the 1-entry in column 1. The set containing these indices is denoted by J w (α). Secondly, the limit shape can be extended for j ∈ [n] J w (α) using a symmetry argument.

Intermediate permutations
Theorem 4.2. Let σ 0 = id and σ k = s w1 · · · s w k for k ∈ [N ], where w = w 1 . . . w N is a sorting network. Let P n be the uniform probability measure on R 132 n , the set of n-element 132-avoiding sorting networks. Finally, let (1) and J c w (α) = [n] J w (α). For all 0 α 1 and > 0, as n → ∞. By symmetry, for all 0 α 1 and > 0, as n → ∞.    (Figure 13b). The shift can be performed by sending y to y + x since y shifted = y non-shifted + x. Thus we get the curve 2x+y √ 2 =g α ( −y √ 2 ) and consequently the explicit limit formula x = 1 Figure 13c. The missing part of the limiting permutation matrix is obtained from the diagram by the symmetry in Proposition 2.3.

Trajectories
The scaled trajectory f i (α) = f w,i (α) of i in an n-element 132-avoiding sorting network w is defined by for αN ∈ Z, and by linear interpolation for other α ∈ [0, 1]. Figure 16 contains some examples. Proof. We show that all elements between i and j have to pass i, and that all elements smaller than i have to pass both i and j before i and j can be swapped. Note that elements k such that i < k < j cannot be swapped with j before they have been swapped with i. Otherwise an intermediate permutation would have the pattern 132. Similarly, all k with k < i have to pass i and hence also j before i and j can be swapped.
In general the element k starts to move when 1 reaches position k, that is, at the smallest t such that the first row of the tableau Q w1...wt (defined above Theorem 2.2) has length k − 1. Figure 16 shows how the trajectories are constant until they are intersected by the trajectory of 1. This time is given by the limit shape of shifted SYT of staircase shape, and is equal to (1 − √ 1 − x 2 )/2, where x = k/n. This is the content of the proposition below.
Proposition 5.2. For n ∈ N let R 132 n denote the set of n-element 132-avoiding sorting networks and P n be the uniform probability measure on R 132 n . Then for all > 0, lim n→∞ P n w ∈ R 132 n : sup The function 1 − √ 1 − x 2 2 is continuous and strictly increasing. Hence its inverse is also continuous, so for every > 0 there exists a δ > 0 such that This proves the claim with f w,1 (α) = j/n and α = T (1, j)/N . Note that since 2 √ α − α 2 = 1 at α = 1 2 , t 0 (α) has to equal 1 for 1 2 < α 1. Note that by symmetry, the limit shape of the trajectory of n can be obtained by the transformation α → 1 − α.
In general, the limit shapes of trajectories are determined by the limit shape of shifted SYT of staircase shape. The lemma below states that we can read the steps at which the element m switches position -and hence the position of m at a given step -by reading the entries of Q w first along the anti-diagonal i + j − 1 = m − 1 and then along the column i = m. Then Proof. This follows from Theorem 4.1 by the observation that the element m can only switch places with elements m < m until m reaches the first column, and that m + 1 cannot pass m before m has, that is, the row m has reached length m − m . This proves the part concerning the anti-diagonal i + j − 1 = m − 1. The remaining part follows by the symmetry in Proposition 2.3.
The next theorem formulates Lemma 5.3 in the limit in terms of the limit shape in Theorem 3.7. The function f β (α) below is the non-constant part of the limit of the trajectory of βn and is determined by the limit surface.

1250
On random shifted SYT and 132-avoiding sorting networks Finally, let R 132 n denote the set of n-element 132-avoiding sorting networks and let P n be the uniform probability measure on R 132 n . Then for all > 0, lim n→∞ P n w ∈ R 132 n : sup Proof. This is simply a shifted and scaled version of Lemma 5.3 together with Theorem 3.7. Note that L(x, y) restricted to D β is continuous by the continuity of L(x, y) and also strictly increasing since D β intersects each level curve exactly once.
Informally, by the result above we can trace the trajectory of βn by following the limit shape along y = β until x = y, and then along x = β. If the height α = L(x , y ) is given by some point (x , y ) along this curve, then the trajectory of βn is at height y − x at time α. Figure 17 illustrates this. Combined with the implicit definition of L(x, y), this means that it is difficult to compute the trajectories of arbitrary elements explicitly. However, certain values are obtainable for all of the trajectories. In the limit, with high probability, the trajectory of βn is at height 1 at Note that by the proof of Lemma 5.3, this is the time α at which the diagonal side of Q → w1...w αN has reached length βn . On the other hand, at time α = 1 2 , the trajectory of βn is at height (1 − 2β)n with high probability. Points from some of the scaled limit trajectories predicted by Theorem 5.4 are plotted in Figure 16 together with the corresponding sampled scaled trajectories. Figure 17. A curve (in red) determining the trajectory of βn , β = 0.4, shown on the right from the limit shape.

Adjacencies
Motivated by the former great circle conjecture [1,Conj. 3] (now [4,Thm. 4]) and trying to understand the geometry of random 132-avoiding sorting networks on the permutahedron, we next study adjacencies. Note that by [1,Thm. 5] and the previous Algebraic Combinatorics, Vol. 3 #6 (2020) two sections, random 132-avoiding sorting networks are not likely to be close to any fixed great circle.
Let w be a reduced word of the longest element w 0 in S n . An index k ∈ [N − 1] is called an adjacency of w if |w k+1 − w k | = 1. Geometrically, adjacencies are special in the following sense.
Proposition 6.1. In the permutahedron, an adjacency k in a reduced word w corresponds to a pair of edges at a π 3 angle. Otherwise the edges corresponding to consecutive letters w k and w k+1 are orthogonal.
Proof. An adjacency corresponds to a pair of adjacent edges of length √ 2 with direction vectors of the form (0, . . . , 0, 1, 0, . . . , 0, −1, 0, . . . , 0) where the indices of either the 1s or the −1s coincide. Hence their scalar product is 1 and the edges constitute an angle of π 3 . In the case of |w k+1 − w k | > 1 the edges corresponding to w k and w k+1 are orthogonal.
Adjacencies in a 132-avoiding sorting network w correspond directly to adjacencies in the SYT Q → w defined above Theorem 2.2 as follows. Let λ dg be a Young diagram. A pair (T, u) of a cell u ∈ λ dg and a SYT T of shape λ dg is called a horizontal adjacency if T (eu) = T (u) + 1. The pair (T, u) is called a vertical adjacency if T (su) = T (u) + 1. An adjacency (T, u) is said to lie in column j of λ dg if u = (i, j). Likewise, in such a case (T, u) is said to lie in row i of λ dg . The definitions are the same for shifted diagrams λ sh . For example, consider the tableau T in Figure 18. Then (T, (1, 1)), (T, (1,2)) and (T, (2,2)) are horizontal adjacencies whereas (T, (2, 3)) is a vertical adjacency. The adjacencies lie in columns 1, 2, 2, and 3 (and rows 1, 1, 2 and 2), respectively.
) is a vertical adjacency. Proof. This follows from the fact that w i is inserted in column w i of Q w1...wi−1 in the Edelman-Greene bijection, and that Q w can be shifted.
Our next goal is to enumerate adjacencies in SYT. To this end define In other words, τ k (T ) exchanges the positions of k and k + 1 if possible, that is, unless they are adjacent in T . See Figure 18. We consider the bijection ∂ k = τ |λ|−1 •· · ·•τ k+1 •τ k from the set of (possibly shifted) SYT of shape λ to itself, where |λ| is the largest entry in an SYT of shape λ. Applying ∂ k to T can be described by the following procedure called partial promotion: (1) Form the sequence of cells u 0 , . . . , u m , called the promotion path, such that • u l+1 = arg min{T (eu l ), T (su l )}. If at some l = m both are undefined, we stop.
(3) Subtract 1 from each label at least k + 1 and insert |λ| into u m . The inverse of ∂ k can be described in a similar way: (1) Form the sequence of cells u m , . . . , u 0 , called the inverse promotion path, such that • u m = T −1 (|λ|), • u l+1 = arg max{T (wu l ), T (nu l )}, • and u 0 is the last possible cell in the sequence such that T (u 0 ) k. (2) Remove the label of u m and slide the entries T (u 0 ) → · · · → T (u m ).
(3) Add 1 to each label at least k. Insert k into u 0 . See Figure 19 for an example. In the case k = 1 partial promotion becomes Schützenberger's promotion [37]. See also [39]. Let A c (T ) be the set of horizontal adjacencies (T, (i, c)) in column c of a (possibly shifted) standard Young tableau T . Let λ be a (possibly shifted) shape and let A c (λ) = ∪ T A c (T ) where T runs over all SYT of shape λ.  Figure 19. Partial promotion ∂ 2 and its inverse, with the promotion paths highlighted. Theorem 6.3. For any column c of a (possibly shifted) shape λ, partial promotion gives a bijection between A c (λ) and the set of SYT of shape λ with largest entry in or after column c + 1. The corresponding result for vertical adjacencies is obtained by replacing "column" with "row".
Proof. Consider a tableau T of shape λ and assume the largest entry |λ| is in column c + 1 or greater. Then the inverse promotion path of τ 1 • τ 2 • · · · • τ |λ|−1 has to cross from column c+1 to c at some unique k(T ), that is, k(T ) is the first index at which the path of ∂ −1 k(T ) = τ k(T ) • · · · • τ |λ|−1 ends in column c of T . By the definition of inverse partial promotion, the choice of k(T ) ensures that k(T ) and k(T ) + 1 are horizontally adjacent with k(T ) in some cell u in column c of ∂ −1 k(T ) (T ). This is illustrated in Figure 20. Hence, by letting φ(T ) = (∂ −1 k(T ) (T ), u) we obtain a map φ : S c (λ) → A c (λ). k k k+1 Figure 20. The bijection in the proof of Theorem 6.3 with k = k(T ) and the path of τ k(T ) • · · · • τ |λ|−1 coloured.
If (S, u) is an adjacency in column c, the largest entry of ∂ S(u) (S) has to be in column c + 1 or greater as there is at least one entry in column c + 1. Hence, by defining ψ(S, u) = ∂ S(u) (S) we obtain a map ψ : A c (λ) → S c (λ).
The proof works for vertical adjacencies as well if one replaces "column" by "row" since promotion and its inverse are symmetric in rows and columns.
Techniques similar to those used in the proof of Theorem 6.3 also appear in [36], for example in the proof of Theorem 1.3 in [36, Section 3]. Theorem 6.3 is true in a greater generality, for example for many skew shapes. The important feature is that every possible promotion path from the maximal entry has to cross column c.
A (shifted) Young diagram is called a (shifted) rectangle if all standard tableaux of its shape contain the largest entry in the same cell. That is, λ = (λ 1 , . . . , λ ) is a shifted rectangle if and only if λ i = λ 1 − i + 1 for all 1 i . Note that shifted staircases are shifted rectangles.
Corollary 6.4. Let λ dg (resp. λ sh ) be a (shifted) rectangle and fix a column c of λ dg (resp. λ sh ). Then the number of horizontal (alt. vertical) adjacencies in column c < λ 1 (alt. row r < ) is equal to f λ (resp. f sh λ ). Proof. This a special case of Theorem 6.3.
The below result, in turn, is a corollary of Corollary 6.4.
Corollary 6.5. The expected number of horizontal (resp. vertical) adjacencies in column c < λ 1 (resp. row r < ) of a uniformly random (shifted) rectangle SYT is equal to 1.
Proof. Apply Corollary 6.4, the definition of expectation, and the fact that each tableau occurs with the same probability. Corollary 6.5 finally yields the following.
Corollary 6.6. The expected number of adjacencies in a random 132-avoiding sorting network of length N is 2(n − 2).
Proof. This is a consequence of Corollary 6.5 by linearity of expectation and Proposition 6.2, since shifted staircases are shifted rectangles.
Compare this with the result of Schilling et al. below. We note that for some (shifted) rectangles the variance of the total number of horizontal (or vertical) adjacencies seems to be an integer.
Conjecture 6.8. Let λ = (n) n . The variance of the number of horizontal (alt. vertical) adjacencies in a SYT of shape λ dg is n − 1. The variance of the number of horizontal (alt. vertical) adjacencies in a shifted SYT of shape ∆ sh n is n − 3. By Proposition 6.2, this would give the variance of the number of i ∈ [n − 1] such that w i+1 = w i + 1 (alternatively, w i+1 = w i − 1) in a random 132-avoiding sorting network w of length N . In the same vein, Reiner has a conjecture [29,Conj. 3] on the variance of the number of i (i + 1) i, 1 i n − 1, in a random sorting network of length N , namely that it is N −4 N −2 for n 4. Some other corollaries of Theorem 6.3 are listed next.
Algebraic Combinatorics, Vol. 3 #6 (2020) Figure 21. The number of adjacencies in an initial segment of a random sorting network of size n = 1000, (left) seems to grow linearly. The number of adjacencies in an initial segment of a random 132avoiding sorting network of size n = 2000 (right) seems to grow like a square root for the first half. As an interesting problem for further study, we would also like to mention the following.
Open problem 6.14. What is the behaviour of |w k − w k+1 | for k ∈ [N − 1] in random sorting networks w of length N ? See Figure 22. For example, if {w k , w k+1 } = {j, j + 2} in a 132-avoiding sorting network, then k and k + 1 are diagonally adjacent in the corresponding shifted SYT Q → w .