On cyclic Schur-positive sets of permutations

We introduce a notion of cyclic Schur-positivity for sets of permutations, which naturally extends the classical notion of Schur-positivity, and it involves the existence of a bijection from permutations to standard Young tableaux that preserves the cyclic descent set. Cyclic Schur-positive sets of permutations are always Schurpositive, but the converse does not hold, as exemplified by inverse descent classes, Knuth classes and conjugacy classes. In this paper we show that certain classes of permutations invariant under either horizontal or vertical rotation are cyclic Schur-positive. The proof unveils a new equidistribution phenomenon of descent sets on permutations, provides affirmative solutions to conjectures by the last two authors and by Adin–Gessel–Reiner– Roichman, and yields new examples of Schur-positive sets. Mathematics Subject Classifications: 05E05 ∗Partially supported by a Bar-Ilan University visiting grant. †Partially supported by Simons Foundation grant #280575. Corresponding author. ‡Partially supported by an MIT-Israel MISTI grant and by the Israel Science Foundation, grant no. 1970/18. the electronic journal of combinatorics 27(2) (2020), #P2.6 https://doi.org/10.37236/8974


Introduction
Let [n] := {1, 2, . . . , n} and let S n denote the symmetric group on [n]. Recall that the descent set of a permutation π ∈ S n is Des(π) := {i ∈ [n − 1] : π(i) > π(i + 1)}. (1) Given any subset A ⊆ S n , we define the quasi-symmetric function where F n,D is Gessel's fundamental quasi-symmetric function [11], defined for D ⊆ [n − 1] by F n,D := A symmetric function is called Schur-positive if all the coefficients in its expansion in the basis of Schur functions are nonnegative. A subset A ⊆ S n is called Schur-positive if Q(A) is symmetric and Schur-positive.
The following long-standing problem was first addressed in [12].
Problem 1. Find Schur-positive subsets of S n .
It is possible to characterize Schur-positive permutation sets using standard Young tableaux (SYT). We write λ/µ n to mean that λ/µ is a skew shape with n boxes, where λ and µ are partitions such that the Young diagram of µ is contained in that of λ. Let SYT(λ/µ) denote the set of standard Young tableaux of shape λ/µ. We draw tableaux in English notation, as in Figure 1. The descent set of T ∈ SYT(λ/µ) is Des(T ) := {i ∈ [n − 1] : i + 1 is in a lower row than i in T }.
For example, the descent set of the SYT in Figure 1  This characterization of Schur-positive sets of permutations is useful because it does not require computing quasisymmetric functions, but rather finding a Des-preserving bijection from permutations to SYT of shapes given by a certain multiset.
In this paper we introduce and study a cyclic analogue of Schur-positive permutation sets, whose definition is motivated by Theorem 2. Before we state Definition 4, we need some background on cyclic descent sets.
The cyclic descent set for rectangular SYT was introduced by Rhoades [15], and extended to some other shapes in [1,10]. This notion was generalized to all skew shapes that are not connected ribbons in [3]. An explicit combinatorial description of cyclic descent sets on SYT(λ/µ) for every skew shape which is not a connected ribbon was recently given by Huang [13]. The following is the main definition in this paper.
where cDes(π) is defined by Eq. (4), cDes(T ) is the cyclic descent set defined in [3,13], and the sum in the RHS is over skew shapes λ/µ that are not connected ribbons (and thus cyclic descent sets exist for SYT of these shapes).
In Theorem 15 we give an alternative characterization of cyclic Schur-positive permutation sets using an invariance property of cyclic descent sets.
It will follow from this characterization that cyclic Schur-positive sets of permutations are always Schur-positive. However, the converse does not hold; in fact, most known examples of Schur-positive sets are not cyclic Schur-positive. One of our goals is to address the following problem. In this paper we present two families of cSp sets of permutations: horizontal rotations of Schur-positive sets, and vertical rotations of inverse descent classes, which include arc permutations. Let us now define these concepts.
To define horizontal and vertical rotations, let c n denote the n-cycle (1, 2, . . . , n) = 23 . . . n1 ∈ S n , and let C n := c n be the cyclic subgroup generated by c n .
Any set A ⊆ S n−1 can be interpreted as a subset of S n by identifying S n−1 with the set of permutations in S n that fix n. With this interpretation, we define the horizontal (respectively, vertical) rotation closure of A ⊆ S n−1 as the set AC n ⊆ S n (respectively, C n A ⊆ S n ). Our first main result (Theorem 23) states that if A ⊆ S n−1 is Schurpositive, then its horizontal rotation closure AC n ⊆ S n is cSp. As a consequence, we show in Corollary 24 that the set of permutations whose inverses have a given number of cyclic descents is cSp.
The inverse of a set of permutations A ⊆ S n is defined as Our second main result (Theorem 25) implies that for any inverse descent class, the distribution of the statistic cDes is the same on its vertical rotations C n D −1 n−1,J as on its horizontal rotations D −1 n−1,J C n . The proof of this result, which settles a stronger version of [9, Conjecture 10.2], involves cDes-preserving operations on grid classes, as defined in [5]. Even though our proof is not bijective in general, we give explicit cDes-preserving bijections between D −1 n−1,J C n and C n D −1 n−1,J when |J| = 1 in Section 5, and when J = [i] in Section 6.
Combining the two main theorems mentioned above, it follows that the set C n D −1 n−1,J of vertical rotations of an inverse descent class is cSp (Theorem 33). In particular, it is Schur-positive, as had been conjectured in [9]. Of special interest is the set A n of arc permutations, which are those permutations in S n where every prefix forms an interval in Z n . As we will see in Section 6, this set is a union of vertical rotations of inverse descent classes, so we deduce (Corollary 42) that it is cSp as well, and we provide a bijective proof of this fact (Theorem 50).

Schur-positive permutation sets
Recall that A ⊆ S n is Schur-positive if the quasi-symmetric function Q(A) from Equation (2) is symmetric and Schur-positive. A direct combinatorial characterization of Schurpositive sets was described in Theorem 2.
Some classical examples of Schur-positive sets of permutations are given in Table 1. Other examples have appeared more recently in [8,9,10].

Cyclic descents for SYT
Originally introduced by Rhoades [15] in the setting of rectangular shapes and later extended to some other shapes in [1,14], the notion of cyclic descent set of SYT was defined for arbitrary skew shapes in [3]. The following definition, introduced in [1,3], was motivated by the basic common properties of cyclic descent sets of permutations, defined in Equation (4), and of SYT in the cases for which the definition was known at that time. For

Schur-positive subset of S n Reference
Related examples Knuth class [11] subsets invariant under Knuth relations (e.g. inverse descent classes, 321-avoiding permutations) conjugacy class [12,Thm. 5.5] subsets invariant under conjugation (e.g. involutions) permutations with a fixed inversion number [4,Prop. 9.5] Table 1: Classical examples of Schur-positive sets of permutations a set D ⊆ [n] and an integer i, we use the notation i + D := {i + d mod n : d ∈ D} ⊆ [n]. Throughout the paper, addition of elements in [n] will be interpreted modulo n. When T is a set of SYT, it will be assumed that the descent map Des is the one defined in Equation (3). Similarly, when T is a set of permutations, Des and cDes will refer to the maps defined in Equations (1) and (4), respectively. An explicit combinatorial description of a cyclic descent extension on SYT(λ/µ) for every skew shape which is not a connected ribbon was recently given by Huang [13].
Example 8. Write µ 1 ⊕ µ 2 ⊕ · · · ⊕ µ t to denote the skew shape consisting of t connected components µ 1 , . . . , µ t , ordered from southwest to northeast. A strip is a shape µ 1 ⊕ µ 2 ⊕ · · · ⊕ µ t , each of whose connected components has either only one row or only one column. In the special case where SYT is a horizontal strip shape with at least 2 components, we have cDes(T ) := {i ∈ [n] : i + 1 mod n is in a lower row than i in T }.
The following lemma is used in Section 3.

Basic examples
Definition 4 introduces the concept of cyclic Schur-positivity (cSp), which is the central notion of this paper. Let us make some remarks about this definition. First, note the analogy with Theorem 2, which characterizes Schur-positivity of sets of permutations. One difference, however, is the use of skew shapes in Definition 4. This modification is needed because SYT of hook shapes do not carry a cyclic descent extension. Second, note that Lemma 10 implies that, for every skew shape λ/µ which is not a connected ribbon, the polynomial is well defined, in the sense that it does not depend on the choice of cDes.
Example 11. Recall the cyclic subgroup C n generated by the n-cycle c n = (1, 2, . . . , n). By the definitions in Example 8, we have thus C n is cSp.
thus S n is cSp.

An alternative characterization
The following equivariance property will be used to give an alternative characterization of cSp sets.
Remark 14. A subset A ⊆ S n is invariant under horizontal rotation if A = Ac n , where c n = (1, 2, . . . , n). Subsets invariant under horizontal rotation are cDes-invariant, since one can simply take ψπ = πc −1 n in Definition 13.
The following characterization of cSp sets will be used in the rest of the paper.

Theorem 15. A subset A ⊆ S n is cyclic Schur-positive if and only if it is Schur-positive and cDes-invariant.
Before proving this theorem, let us consider some examples of its usage.
Thus, by Theorem 2, it is Schur-positive. On the other hand, by Remark 14, C n is cDes-invariant. Hence, by Theorem 15, C n is cSp, in agreement with Example 11.
Remark 18. By Theorem 15, every cSp set of permutations is Schur-positive. The converse does not hold, as Example 17 shows.
We now turn our attention to the proof of Theorem 15. In the next two lemmas, we assume that A ⊆ S n is Schur-positive and cDes-invariant, and we let (b λ ) λ n be nonnegative integers such that which exist by Theorem 2.
Lemma 19. For every 0 k < n, Proof. For every 0 k < n, there is a unique SYT T with Des(T ) = [k], namely the tableau of shape (n − k, 1 k ) having 1, . . . , k + 1 in the first column and 1, k + 2, . . . , n in the first row. Comparing the coefficients of t [k] on both sides of Equation (10) completes the proof.
Lemma 20. For every 0 k < n, the alternating sum Proof. To simplify notation, let us first write, for any S ⊆ [n], Combining these facts we see that The telescoping sum on the right-hand side further reduces to using that A [n] = ∅ by the non-Escher property. This expression is clearly nonnegative, and it equals zero when k = 0, since A [0] = ∅ again by the non-Escher property.
Lemma 21. Let A ⊆ S n be cDes-invariant. Assume that there are nonnegative constants Further, assume that b λ/µ = 0 whenever λ/µ is a connected ribbon. Then A is cSp.
Proof. Let B(t) be the quantity in Equation (11). As no ribbon shape appears with positive multiplicity on the right-hand side, we know from Theorem 7 that a cyclic descent extension (cDes, ψ) exists so that where cDes is now the cyclic descent for permutations given by Equation (4).
the electronic journal of combinatorics 27(2) (2020), #P2.6 By definition, this cyclic descent for permutations satisfies the extension and non-Escher properties from Definition 6. In addition, since A is cDes-invariant, there exists a bijection ρ : A → A through which cDes satisfies the equivariance property. Thus, the pair (cDes, ρ) is a cyclic descent extension on A.
For every subset ∅ = J = {j 1 < · · · < j t } ⊆ [n], let b(J), c(J) and d(J) be the coefficients of t J in B(t), C(t) and D(t), respectively. It follows from Lemma 10 that Thus C(t) = D(t), which implies that A is cSp.
Proof of Theorem 15. For 0 k n, we have Observe that none of the shapes on the right-hand side are ribbon shapes. It now follows from Lemma 21 that A is cSp.
For the converse, assume now that A is cSp. Setting t n = 1 in Equation (5) gives Applying the vector space isomorphism from the multilinear subspace of the formal power series ring Z[t 1 , t 2 , . . .] to the ring of quasisymmetric functions, defined by t J → F n,J , we get Finally, by Equation (5), A is cDes-invariant because so is the corresponding collection of SYT of the skew shapes given by the right-hand side.

Horizontal rotations
We conclude this section with some applications of Theorem 15 to sets of permutations that are invariant under horizontal rotation. The next result follows immediately from Theorem 15 together with Remark 14.
Theorem 22. If A ⊆ S n is Schur-positive and invariant under horizontal rotation, then it is cSp.
Another consequence is the fact that horizontal rotation closures of Schur-positive sets are cSp.
Theorem 23. For every Schur-positive set A ⊆ S n−1 , the set AC n ⊆ S n is cSp.
Proof of Theorem 23. For every Schur-positive set A ⊆ S n−1 , the set AC n ⊆ S n is Schurpositive by [10, Theorem 1.1]. Since AC n is invariant under horizontal rotation, Theorem 22 implies that it is cSp.
Corollary 24. For every n > k 1, the set Proof. It is shown in [9, Corollary 7.7] that C n,k is Schur-positive. By [9, Lemma 6.4], it is invariant under horizontal rotation. Thus, by Theorem 22, it is cSp.
A more transparent, self-contained proof of Corollary 24 will be given in Section 4.2.

Vertical versus horizontal rotations
In this section we prove the following equidistribution result, and we discuss applications of it. Recall the definition of D n−1,J from Equation (6).
Our proof is not bijective, but explicit bijections for the special cases when J = {j} (singletons) and J = [i] (prefixes) will be given in Sections 5 and 6, respectively.

Proof of Theorem 25
We first recall some basic definitions. A composition (respectively, weak composition) of n 0 is a finite sequence of positive (respectively, nonnegative) integers γ = (γ 1 , . . . , γ t ) whose sum is n.
For the remainder of this section, fix a composition γ = (γ 1 , . . . , γ t ) of n and denote by J ⊆ [n − 1] its corresponding subset. The relevance of these definitions to the theorem at hand is the fact that I⊆J C n D −1 n−1,I = C n S(γ 1 , . . . , γ * t ) and As we shall see, it is easier to first argue in terms of such unions and then, via an application of inclusion-exclusion, conclude our desired result. To keep track of the position of the largest letter, we introduce two further refinements. Let so that V k γ is the result of vertically rotating the permutations in S(γ 1 , . . . , γ * t ) until the largest value is in the kth position. Likewise, define so that this set is the result of horizontally rotating the same set of permutations until the largest letter is again in the kth position. We can now state our main technical lemma from which the proof of Theorem 25 follows almost immediately.
Lemma 26. For every k n, there exists a cDes-preserving bijection between V k γ and H k γ .
Assuming this lemma for the moment, let us prove the theorem.
Proof of Theorem 25. By Lemma 26, it follows that With J ⊆ [n − 1] defined as above, it follows from (14) and the principle of inclusionexclusion that Additionally, the analogous equality involving right multiplication by C n also holds. Together, these facts yield the desired statement.
To prove Lemma 26, we recall the notion of a shuffle. Let ι n := 12 . . . n denote the increasing permutation. For any nonnegative integers a and b, define to be the set of permutations in S a+b where the letters in [a] appear from left to right in increasing order, and so do the letters in [a + b] \ [a]. By definition, The next lemma belongs to mathematical folklore, but we include a proof for the sake of completeness, as well as some examples.
Lemma 27. For every a, b > 0, there exists a Des-preserving bijection the electronic journal of combinatorics 27(2) (2020), #P2.6 Proof. Permutations π ∈ ι a ¡ ι b can be encoded bijectively as words w over a binary alphabet {1, 2} with a 1s and b 2s, where w i = 1 if π i a, and w i = 2 if π i > a.
Given π ∈ ι a ¡ι b , let w its corresponding binary word. By splitting w at the descents of π, which correspond to occurrences of 21 in w, we can write and define ϕ(π) to be the permutation in ι b ¡ ι a encoded by f (w). By construction, Des(ϕ(π)) = Des(π).
It is possible to extend the construction in Lemma 27 to shuffles of t 2 increasing sequences. Let a 1 , . . . , a t , n be positive integers such that a 1 + · · · + a t = n. We can explicitly construct a bijection ϕ : ι a 1 ¡ · · · ¡ ι at → ι at ¡ · · · ¡ ι a 1 as follows.
For a given word u over the alphabet {1, . . . , t} and 1 j < t, define f j (u) to be the word obtained by fixing in place all the entries of u that are not equal to j or j + 1, and applying the map f from the second proof of Lemma 27 to the subword of u consisting of the entries j and j + 1 (ignoring the other entries).
Given π ∈ ι a 1 ¡ · · · ¡ ι at , we can encode it as a word w of length n over the alphabet {1, . . . , t} with a j js for 1 j t, by letting w i = j if a 1 + · · · + a j−1 < π i a 1 + · · · + a j , for all i.
Refining this set by the value in the kth position, we get To obtain our claim, vertically rotate each permutation in (17) so that the value in the kth position is largest, and then horizontally rotate the resulting permutation k positions to the left.
The decomposition in the previous lemma motivates the next result.
Armed with these technical lemmas, we prove Lemma 26.
Applying Lemma 27 to the right-hand side and noting that all permutations end with the largest letter n, we get a cDes-preserving bijection between this set and S(γ 1 , . . . , γ * t ). This completes our proof.

Consequences of Theorem 25
The following is one of the main applications of the theorem. Proof. As shown in [11] (see also Table 1), inverse descent classes are Schur-positive. Hence, by Theorem 23, for every J ⊆ [n − 2], the horizontally rotated inverse descent class D −1 n−1,J C n is cSp. Finally, by Theorem 25, the distribution of cDes on C n D −1 n−1,J is the same as on D −1 n−1,J C n , completing the proof.
Next we apply Theorem 33 to give a self-contained proof of Corollary 24, which states that the set of permutations with a given inverse cyclic descent number is cSp.
The set C n,k had been shown to be Schur-positive in [9,Corollary 7.7]. A significantly stronger version of this result is conjectured in [2, Conjecture 7.1]. Originally formulated in the language of cyclic quasi-symmetric functions, cyclic compositions and cyclic permutations, this conjecture equivalent to the statement that, for every ∅ J [n], the set of permutations in S n whose inverse has cyclic descent set of the form i + J for some i (with addition modulo n as usual) is Schur-positive. The following consequence of Theorem 33 provides an affirmative solution to this conjecture.
Proof. For j ∈ J, let ∆ j := D n−1,−j+(J\{j}) , viewed as a subset of S n , and let A denote the set in Equation (21). Given π ∈ S n , we have that cDes(π) = J if and only if there exists j ∈ J such that j − 1 / ∈ J and π = σc j n for some σ ∈ ∆ j . Indeed, the 'if' direction is clear, and the other is obtained by taking j = π −1 (n). It follows that cDes(π) = i + J if and only if there exists j ∈ J such that j − 1 / ∈ J and π = σc j+i n for some σ ∈ ∆ j . Letting i vary, we get Any two sets in this union are either equal or disjoint. Specifically, ∆ j 1 C n ∩ ∆ j 2 C n = ∅ unless −j 1 + J = −j 2 + J, in which case ∆ j 1 C n = ∆ j 2 C n . It follows that we can express A −1 as a disjoint union of sets ∆ j C n for j in some subset J ⊆ J. Taking inverses, we get which is cSp by Theorem 33, and thus Schur-positive by Theorem 15.

An explicit bijection for shuffles of two increasing sequences
In this section and the next one we describe explicit bijections proving Theorem 25 in two special cases. This section deals with the case that |J| = 1, and Section 6 will deal with the case J = [i].
In the rest of this section, suppose that J contains one element, that is, J = {j}. In this case, D −1 n−1,J consists of shuffles of two increasing sequences of fixed length. Specifically, D −1 n−1,J = S(j, (n − j) * ) \ {12 . . . n}. We give an explicit bijection Ψ : C n D −1 n−1,{j} → D −1 n−1,{j} C n that preserves cDes and the position of n. This bijection is obtained by analyzing the proof of Theorem 25, and using the map ϕ in the proof of Lemma 27 to make the bijection explicit. For a word w over the alphabet {1, 2}, recall from the proof of Lemma 27 that f (w) is obtained by keeping the descents 21 untouched, and changing each consecutive block of letters of w that is not part of a 21, which must be of the form 1 r 2 s , into 1 s 2 r . Additionally, for p < q, define f [p,q] (w) to be the word obtained by applying the above operation f to the factor w p w p+1 . . . w q , and keeping the other entries of w unchanged.
Next we describe how to obtain Ψ(π) for given π ∈ C n D −1 n−1,{j} . Let k = π −1 (n) be the position of n. If k = n, we simply define Ψ(π) = π. Suppose now that k = n. Rotate π horizontally k positions to the left, so that its rightmost entry becomes n. The resulting permutation πc k n is a shuffle of two increasing sequences, so it can be encoded as a word w over {1, 2} by placing 1s in the positions corresponding to the lower increasing sequence, and 2s elsewhere. If w n−k = 1, let w = f [n−k+1,n−1] (w), else (that is, if w n−k = 2) let w = f [1,n−k−1] (w). Let w = f [1,n−1] (w ), and let σ be the permutation in D −1 n−1,{j} (viewed as a subset of S n ) encoded by w . Let Ψ(π) = σc −k n , that is, the permutation obtained by rotating σ horizontally k positions to the right. A schematic description of this construction is given in Figure 3.

Arc permutations
In this section we give an explicit bijective proof of Theorem 25 in the case J = [i]. Unlike the bijection Ψ from Section 5, this bijection does not follow from our proof of Theorem 25. We will see that the theorem, in this special case, becomes a statement about arc permutations.
Definition 40. A permutation π ∈ S n is an arc permutation if, for every 1 j n, the first j letters in π form an interval in Z n . Denote by A n the set of arc permutations in S n . Figure 3: A schematic description of the map Ψ. Permutations π ∈ C n D −1 n−1,{j} (obtained by vertically rotating D −1 n−1,{j} ) with π(k) = n fall into two cases. Here α 1 + β 1 = j, α 2 + β 2 = n − j, and β 1 + β 2 = k.
Other combinatorial properties of these permutations, such as their descent set distribution, are studied in [8]. In particular, it follows from [8,Theorem 5] that A n is a Schur-positive set. Notice that while c n A n = A n for every n, we have that A n c n = A n for n > 3. In other words, A n is invariant under vertical rotation but not under horizontal rotation, and thus Theorem 22 does not apply. However, we will show that it is possible to express arc permutations in terms of vertical rotations of inverse descent classes.

Permutation classes and grids
Definition 41. A sequence of integers a 1 , . . . , a n is • left-unimodal if it is a union of an increasing subsequence and a decreasing subsequence, which intersect at the first letter a 1 ; • right-unimodal if the sequence a n , a n−1 , . . . , a 1 is left-unimodal.
We say that a permutation π ∈ S n has one of the above properties if the sequence π(1), π(2), . . . , π(n) does. Denote the sets of left-unimodal and right-unimodal permutations in S n by L n and R n , respectively. Note that for every n 1, π ∈ L n (respectively, π ∈ R n ) if and only if, for every 1 j n, the first (respectively, last) j letters in π form an interval in Z. These sets can be described in terms of pattern avoidance as L n = S n (132, 312) and R n = S n (231, 213), respectively.
Arc permutations are obtained as vertical rotations of left-unimodal permutations, that is, A n = C n L n−1 . Using that we can express A n as a disjoint union of vertically rotated inverse descent classes: The next result follows now from Theorem 33.
Corollary 42. The set A n is cSp.
the electronic journal of combinatorics 27(2) (2020), #P2.6 Next we recall the notion of geometric grid classes from [5], which will be useful for our bijection in Subsection 6.2.
Definition 43. A geometric grid class consists of those permutations that can be drawn on a specified set of line segments of slope ±1, arranged according to the positions of the nonzero entries of a matrix M with entries in {0, +1, −1}. Specifically, a permutation π ∈ S n can be drawn on these line segments if n points can be placed on them so that the ith point from the left is the π(i)th point from the bottom, for all 1 i n.
Example 44. Left-unimodal and right-unimodal permutations are those in the geometric grid classes of the matrices respectively. A drawing of a left-unimodal permutation on the corresponding grid is shown in Figure 4. Being vertical rotations of left-unimodal permutations, arc permutations are precisely those that can be drawn on one of the grids in Figure 5, which are obtained by vertically rotating the grid in Figure 4. This pictorial grid description implies the following property. Observation 45. A permutation π ∈ S n with π(j) = n is an arc permutation if and only if one of the following holds: • the prefix π(1), . . . , π(j − 1) is a left-unimodal sequence on [j], and the suffix π(j + 1), . . . , π(n) is a right-unimodal sequence on [n − 1] \ [j], or • the prefix π(1), . . . , π(j − 1) is a left-unimodal sequence on [n − 1] \ [n − j], and the suffix π(j + 1), . . . , π(n) is a right-unimodal sequence on [n − j].

A cDes-preserving bijection
Next we give a bijection between D −1 n−1,[i] C n and C n D −1 n−1,[i] that preserves the cyclic descent set. By allowing to i range between 0 and n − 2 and using Equations (22) and (23), we obtain a bijection between L n−1 C n and A n . In fact, we will describe the bijection in this setting.
A key fact used in the construction below is that the descent set map Des is a bijection between L n−1 and the power set 2 [n−2] .
Definition 46. Let φ : L n−1 C n −→ A n be the following map. Given π ∈ L n−1 C n , let j = π −1 (n) be the position of the letter n in π, and construct φπ as follows: • Let (φπ)(j) = n.
• If n ∈ cDes(π), then the set of the leftmost j − 1 entries in φπ is equal to • The order of the first j − 1 entries in φπ is given by the only left-unimodal permutation having descent set Des(π) ∩ [j − 2].
• The order of the last n − j entries in φπ is given by the only right-unimodal permutation having descent set {i − j : i ∈ Des(π), i > j}.
Proof. By construction, cDes(φπ) = cDes(π), and by Observation 45, φπ ∈ A n . To prove that φ is a bijection, it suffices to describe its inverse map. Define ψ : A n −→ L n−1 C n as follows. Given σ ∈ A n , let j = σ −1 (n) be the position of n.
It is easy to verify that ψ and φ are inverses of each other.

From arc permutations to SYT
Next we give a bijective proof of Corollary 42. The proof, which relies on the bijection from Section 6.2, provides an explicit set of SYT on which cDes has the same distribution as it has on arc permutations (as in Definition 4).
Proof. Using Equation (23), it suffices to show that for every 0 k < n − 1, The left equality follows from Theorem 25. To prove the right equality, we construct a cDes-preserving bijection f between D −1 n−1,[k] C n and SYT((n − k − 1, 1 k ) ⊕ (1)). Given σ ∈ D −1 n−1,[k] C n , write σ = τ c −j n , where τ ∈ D −1 n−1,[k] ⊆ S n and 0 j < n. Let f (τ ) be the SYT of shape (n − k − 1, 1 k ) ⊕ (1) having entry n in the northeast component, and whose first column consists of the entry 1 and the elements in the set 1 + Des(τ ). Then let f (σ) = j + f (τ ), with addition defined as in Example 9. Finally, apply the definition of cDes in Equation (7) to verify that f is a cDes-preserving bijection.
Composing the above bijection f with the bijection φ −1 : A n → L n−1 C n from Definition 46, we obtain a cDes-preserving bijection from A n to the set n−2 k=0 SYT((n − k − 1, 1 k ) ⊕ (1)). An example is shown in Figure 8.