Fixed Points of the Evacuation of Maximal Chains on Fuss Shapes

For a partition λ of an integer, we associate λ with a slender poset P the Hasse diagram of which resembles the Ferrers diagram of λ. Let X be the set of maximal chains of P . We consider Stanley’s involution ε : X → X, which is extended from Schützenberger’s evacuation on linear extensions of a finite poset. We present an explicit characterization of the fixed points of the map ε : X → X when λ is a stretched staircase or a rectangular shape. Unexpectedly, the fixed points have a nice structure, i.e., a fixed point can be decomposed in half into two chains such that the first half and the second half are the evacuation of each other. As a consequence, we prove anew Stembridge’s q = −1 phenomenon for the maximal chains of P under the involution ε for the restricted shapes.


Introduction 1.Schützenberger's evacuation
Promotion and evacuation are bijections on the set of linear extensions of a finite poset.It is well known that the RSK algorithm establishes a bijection between the permutations the electronic journal of combinatorics 25(1) (2018), #P1.33 of {1, 2, . . ., n} and ordered pairs of n-cell standard Young tableaux of the same shape [10, pp. 320-321].Evacuation was originally devised by Schützenberger to describe this bijection without involving the RSK algorithm [6].Later Schützenberger extended the definition of evacuation to the linear extensions of a finite poset, described in terms of an operation called promotion [7].One of the fundamental properties Schützenberger proved is that the evacuation is an involution.
Schützenbeger's work was simplified by Haiman [3], whose idea is to express linear extensions as words and then define the promotion and evacuation in terms of elementary operators on these words.For a finite poset P of p elements, a linear extension f : P → {1, . . ., p} of P can be expressed as the word u 1 u 2 . . .u p , where u i = f −1 (i) ∈ P for 1 i p.Let L(P ) be the set of linear extensions.For 1 i p − 1, define operators τ i : L(P ) → L(P ) by τ i (u 1 u 2 . . .u p ) = u 1 u 2 . . .u p , if u i and u i+1 are comparable in P u 1 u 2 . . .u i+1 u i . . .u p , otherwise.

Stanley's point of view
Stanley noticed that the properties of promotion and evacuation depend only on the relations of τ i 's defined in Eq. ( 1) and hence the theory of promotion and evacuation can be extended to a more general context.It is known that the set J(P ) of all order ideals of P , ordered by inclusion, is a finite distributive lattice of rank p and that there is a bijection between the maximal chains ∅ = I 0 ⊂ I 1 ⊂ • • • ⊂ I p = P of J(P ) and the linear extensions of P [9, §3.5], associated with this chain the linear extension f : P → {1, . . ., p} defined by f (t) = i if t ∈ I i − I i−1 .Moreover, every interval of rank 2 of J(P ) contains either three or four elements.Stanley [8] described the promotion and evacuation on maximal chains of J(P ) by extending the definition of τ i 's as follows.For a maximal chain C : ∅ = I 0 ⊂ I 1 ⊂ • • • ⊂ I p = P of J(P ), either the interval [I i−1 , I i+1 ] contains the three elements I i−1 , I i , I i+1 or there is exactly one other element I ′ in this interval, i.e., I i−1 ⊂ I ′ ⊂ I i+1 .In the former case define Cτ i = C; in the latter case Cτ i is obtained from C by replacing I i with I ′ .
As pointed out by Stanley, the same definition of τ i works for any finite graded poset with a unique minimal element 0, unique maximal element 1 and the property that every interval of rank 2 contains either three or four elements, called slender posets.He also mentioned some examples of slender posets, such as intervals in the Bruhat order of Coxeter groups and face posets of regular CW-spheres.

Our work
In this paper we consider some families of slender posets the (tilted) Hasse diagrams of which resemble Ferrers diagrams of partitions of an integer, elaborate the properties of the evacuation of maximal chains of the posets and characterize the maximal chains fixed under evacuation.
For a partition λ = (m 1 , . . ., m k ) of n, denoted by λ ⊢ n, we associate λ with a graded poset (P, ) on a P of lattice points in the plane Z × Z define as follows (sometimes we denote the relation by P when there is a possibility of confusion).
(iii) The Hasse diagram of (P, ) comprises n unit squares in the form of Ferrers diagram of λ.
(0, 0) A maximal chain of (P, ) forms a lattice path from 0 to 1 using north step (1, 0) and east step (0, 1) staying within the Hasse diagram.Let N and E denote a north step and an east step, respectively.Let p be the rank of (P, ) and let X be the set of maximal chains of (P, ).For convenience, members of X are written as words on the alphabet {N, E}.For a maximal chain C = z 1 • • • z p ∈ X with z j ∈ {N, E} (1 j p), the evacuation of C is another maximal chain in X , denoted by Cϵ.The elementary operators τ i : X → X (1 i p − 1) that generate evacuation ϵ can be equivalently defined as follows.The chain Cτ i is obtained from C by interchanging the steps z i and z i+1 if the resulting chain remains to be a member of X ; and Cτ i = C otherwise.For example, for the partition λ = (2, 1) ⊢ 3, the associated poset (P, ) is of rank 4 with three elementary operators τ 1 , τ 2 , τ 3 (see Figure 1) and the operator ϵ = τ 1 τ 2 τ 3 • τ 1 τ 2 • τ 1 .For the maximal chain C = NEEN of (P, ), the evacuation of C, Cϵ = NNEE, is obtained through the process shown in Figure 2.
By a fundamental property of evacuation obtained by Schützenberger [6,7] (see also the proof given by Stanley [8, Lemma 2.2]), the operator ϵ establishes an involution on X .The main result in this paper is that we obtain an explicit characterization of the fixed the electronic journal of combinatorics 25(1) (2018), #P1.33 The process of evacuation of a maximal chain of the poset for λ = (2, 1).
points of the map ϵ : X → X when λ = (n s , (n − 1) s , . . ., 1 s ) is a stretched staircase for positive integers s, n or λ is a rectangular shape (Theorem 5.4 and Theorem 6.1).As a consequence, we prove anew Stembridge's q = −1 phenomenon for the maximal chains of (P, ) under the involution ϵ for the restricted shapes.

Cyclic sieving phenomenon
The cyclic sieving phenomenon is an enumerative property that the orbit structures of a cyclic action on a set X of combinatorial objects are encoded in an enumerator of the set X.More precisely, a triple (X, X(q), ⟨c⟩) consisting of a finite set X, a polynomial X(q) ∈ Z[q], and a cyclic group ⟨c⟩, generated by an element c of order n, acting on X is said to exhibit the cyclic sieving phenomenon (CSP) if for all integers d, the number of elements fixed by c d equals the evaluation is the root of unity of order n.The CSP was first defined by Reiner, Stanton and White [4].The special case when ⟨c⟩ has order 2 was also known as Stembridge's q = −1 phenomenon.
Stanley presented an instance of CSP for the linear extensions of a finite poset P under evacuation.For a linear extension ω = u 1 • • • u p ∈ L(P ), the descent set Des(ω) of ω is defined by Des(ω) = {i : u i > u i+1 , 1 i p}.The CSP involves the enumerator W (q) of L(P ) respecting the comajor index comaj(ω), where comaj(ω) = i∈Des(ω) (p − i).He proved that W (−1) coincides with the number of self-evacuating linear extensions of P , i.e., ωϵ = ω, making use of another family of linear extensions called domino linear extensions as the intermediate stage [8,Theorem 3.1].For the poset (P, ) associated with λ = (n s , (n − 1) s , . . ., 1 s ) or λ = (n sn ), we give an alternative proof of the CSP result in terms of the maximal chains of (P, ) under the action of evacuation ⟨ϵ⟩.
As a q-polynomial for our CSP, we consider the enumerator of the maximal chains of (P, ) with respect to the statistic area, the number of unit squares above a maximal chain the electronic journal of combinatorics 25(1) (2018), #P1.33 C ∈ X within the Hasse diagram of (P, ).Let X(q) = C∈X q area(C) .For example, with the partition λ = (2, 1) ⊢ 3, the associated poset contains five maximal chains, shown in Figure 3, with area-enumerator X(q) = 1 + q + 2q 2 + q 3 .Note that X(−1) = 1 and there is exactly one maximal chain fixed by ϵ.Hence (X , X(q), ⟨ϵ⟩) exhibits CSP.However, the map ϵ does not necessarily reverse the parity of the statistic area.As shown in Figure 3, the maximal chains in each orbit have area of the same parity.
With the area-enumerator X(q), a partition λ of an integer is called a good shape if the triple (X , X(q), ⟨ϵ⟩) of the poset associated with λ exhibits CSP.One can check that the partition λ = (2, 2, 1) ⊢ 5 is not a good shape.A natural question is that what kind of partitions is a good shape?In the context of Coxeter combinatorics, there are two families of fundamental shapes, namely, Fuss shapes of type A and type B. A Fuss shape of type A is a stretched staircase defined by the partition λ = (n s , (n − 1) s , . . ., 1 s ) for positive integers s, n, and a Fuss shape of type B a rectangular shape defined by the partition λ = (n sn ).As a consequence of the main result, Fuss shapes of type A and rectangular shapes, including Fuss shapes of type B, are good shapes.

The structure of this paper
The proof for Fuss shapes of type A occupies a large portion of this paper.In section 2, we evaluate X(q) at q = −1.Since X(q) has no closed form, the evaluation makes use of the generating function of an alternative expression of X(q).Sections 3, 4 and 5 are devoted to characterize and enumerate the fixed points of the map ϵ : X → X .The characterization of the fixed points is quite neat but the proof is relatively sophisticated.Subject to a parity-condition, the maximal chains C ∈ X fixed by evacuation can be factorized in half as C = C 1 C 2 such that C 2 is the evacuation of C 1 and vice versa.Some interesting and crucial points of the proof are listed below.
(i) We discover an interesting factor-swapping property of the evacuation of C ∈ X (see Proposition 3.7), which leads to a factorization of C into building blocks.
(ii) We come up with the notion of primitive factorization of C ∈ X , which enables a characterization of Cϵ (see Theorem 4.5).
(iii) The characterization of the evacuation of primitive blocks in Proposition 4.3 is critical, which enables the determination of the primitive blocks fixed by evacuation (see Proposition 5.1) and the fixed points of the map ϵ : X → X (see Theorem 5.4).
The proof for rectangular shapes is given in section 6. Concluding remarks and some problems for further studies are given in section 7.
2 Evaluation of X(−1) for posets of stretched staircases For positive integers s and n, the Fuss-Catalan number counts the number of lattice paths, called s-Dyck paths of width n, from the origin (0, 0) to the point (n, sn) using N and E steps staying weakly above the line y = sx.When s = 1 they are ordinary Dyck paths.Let F (s) n be the set of s-Dyck paths of width n.We consider the enumerator of the paths π ∈ F (s) n with respect to the number α(π) of unit squares enclosed by π and the line y = sx.Define The case s = 1, f n (q), was considered by Carlitz and Riordan [1], and Fürlinger and Hofbauer [2].There is no known explicit form for f (s) n (q).For two integers m < n, let [m, n] = {m, m + 1, . . ., n}.For λ = (n s , (n − 1) s , . . ., 1 s ) ⊢ s n+1 2 , let (P n , ) denote the poset associated with λ defined on the set of points Note that (P n , ) is of rank (s + 1)n.A maximal chain of (P n , ) forms a lattice path from the origin to the point (n, sn) using N and E steps staying weakly above the line y = sx − s.Let X (s) n denote the set of all maximal chains of (P since a maximal chain of (P ) is simply a s-Dyck path of width n + 1 with the initial s steps and the terminal step removed.Sometimes members of X (s) n are also called truncated s-Dyck paths of width n.
Let p = (s + 1)n, the rank of denote the operator of evacuation on X (s) n , which is defined as Let ⟨ϵ n ⟩ be the group of order 2 generated by ϵ n .The CSP result is stated as follows.
Theorem 2.1.For positive integers s and n, let (P n be the set of maximal chains of the poset (P (s) n , ).Let X(q) be the polynomial defined in Eq. ( 4).Let the group ⟨ϵ n ⟩, generated by the operator ϵ n of evacuation, act on X (s) n .Then (X (s) n , X(q), ⟨ϵ n ⟩) exhibits the cyclic sieving phenomenon.

Evaluation of X(−1)
Define the generating function for {f Lemma 2.2.The polynomial F (x, q) satisfies the equation n , there is a standard factorization of π into s-Dyck paths π 1 , • • • , π s+1 , with respect to the first east step E returning to the line y = sx, as , where N i is the last north step before E going from the line y = sx + i − 1 to the line y = sx + i for 1 i s.We observe that The assertion follows from multiplying the equation by x n and summing over n 0.
A s-ballot path is a lattice path from the origin to some point above the line y = sx using N and E steps staying weakly above the line y = sx.The enumeration of the following s-ballot paths will be useful for the evaluation X(−1) and the enumeration of maximal chains of (P (s) n , ) fixed by the operator ϵ n .Proposition 2.3.For any nonnegative integer h, the number of s-ballot paths from the original to the point (n, sn + h) is ), which is the generating function for the number of s-Dyck paths of width n 0. By Lemma 2.2, G satisfies the equation Let r n;h be the number of s-ballot paths from the original to the point (n, sn + h).By a standard factorization, such a path can be factorized into s-Dyck paths π 0 , . . ., π h as , where N i is the last north step from the line y = sx + i − 1 to the line y = sx + i for 1 i h.By an argument similar to the proof of Lemma 2.2, we observe that the generating function for as required.
(i) For s odd, if n is even (ii) For s even, if n is odd.
Proof.First, we evaluate f (s) . Let P = F (x, −1) and Q = F (−x, −1).We discuss the evaluation according to the parity of s.
Case I.For s odd, say s = 2t + 1.By Lemma 2.2, we have , consisting only of the even degree terms.Comparing this equation with Eq. ( 6), we have otherwise.By the proof of Proposition 2.3, for n even we have .
By Eq. ( 4), the assertion (i) follows from X(−1) = (−1) the electronic journal of combinatorics 25(1) (2018), #P1.33 Case II.For s even, say s = 2t.By Lemma 2.2, we have , consisting only of the even degree terms.Comparing this equation with Eq. ( 6), we have Moreover, multiplying both sides of Q = 1 − xP t Q t+1 by P , we have if n is odd.
3 Evacuation on maximal chains of (P (s) n , ) In this section we analyze the behavior of the maximal chains C ∈ X n of (P n , ) under evacuation.We found that the operator ϵ n can be decomposed in a way depending on C such that the maximal chain Cϵ n has a factor-swapping property (see Lemma 3.6 and Proposition 3.7).
Lemma 3.1.We have Proof.By the definition of ϵ n in Eq. ( 5), the electronic journal of combinatorics 25(1) (2018), #P1.33 For convenience, members of X (s) n are also referred to as paths, with north steps and east steps.Given a path C ∈ X (s) n , let E 1 , E 2 , . . ., E n be the east steps of C from left to right.The east step E k is also said to be in the kth column.Let y(E k ) denote the y-coordinate of the endpoint of for 1 k n, which indicates the vertical depth of the endpoint of E k from the line y = sx.Note that 0 d k s (resp.d k < 0) if the endpoint of E k is weakly under (resp.strictly above) the line y = sx.For example, with s = 3 and n = 3, the first path shown in Figure 4 is encoded (1, −1, 2).
We have the following observation about the operators δ 1 , . . ., δ p−1 applied to the paths in X (s) n .Sometimes we write N ℓ for a consecutive ℓ north steps.
Then the following results hold.
and τ i interchanges the ith and (i + 1)th steps of a path C ∈ X (s) n .Since z 1 = E, the path Cδ p−1 is obtained from C by moving z 1 all the way to the end.As a result, the segment z 2 • • • z p of C is moved to the left by one column and hence the code Π(Cδ p−1 ) is obtained as asserted.
(ii) Note that δ * p−1 = τ p−1 • • • τ 1 is the reverse operation of δ p−1 .Since z p = E, we observe that the last step can be moved all the way to the first position subject to the condition d j 0 for all j n − 1.As a result, the segment We consider two restrictions on the depth-codes of the paths C ∈ X (s) n .These restrictions will be used in the factorization of C with factor-swapping property.For 0 ℓ s, let A n (ℓ) ⊆ X Proof.The case ℓ = 0 is trivial since n .For ℓ > 0 and a path C ∈ A n (ℓ), notice that the last ℓ steps of C are north and that they remain unaffected under the operations of , where D is the resulting path of the first p − ℓ steps.Then applying δ * p−ℓ , . . ., δ * p−1 to DN ℓ , by Lemma 3.2(iv) we have For any positive integer m < n, if d m 0 then C can be factorized into two paths We define an operator γ n;m that swaps C 1 with C 2 , under certain restriction.Define In fact, γ n;m appears in a decomposition of the operator ϵ n (see Lemma 3.6).
Then the depth-code of C 1 can be expressed as for 1 j m.Let γ n:m be written as Lemma 3.6.For any positive integer m < n, the operator ϵ n can be decomposed as Proof.For convenience, let q := (s + 1)m and r := p − q = (s + 1)(n − m).Following the relations (1), the operator ϵ n is rearranged as follows.Note that It suffices to consider the initial factor δ p−1 • • • δ r of ϵ n .Let δ r be fixed.From right to left, move the τ 1 of δ r+1 to the right of the τ 2 of δ r+2 .Then move this τ 1 , along with the factor τ 1 τ 2 of δ r+2 , to the right of the τ 3 of δ r+3 .Repeat this process, moving the factor (τ ) to the right of the τ j of δ r+j for all j q − 1.Now, we have the initial factor (τ The stages of the operation is given below.
as required.
the electronic journal of combinatorics 25(1) (2018), #P1.33 On the basis of Lemmas 3.3, 3.4 and 3.6, we have the following fundamental property of evacuation.With this property, the evacuation of the maximal chains of (P (s) n , ) can be factorized into building blocks.
Then the following properties hold.
Proof.Note that C 1 ∈ A m (ℓ) and C 2 ∈ B n−m (ℓ).We compute the evacuation Cϵ n using the decomposition ϵ n = ϵ m γ n;m ϵ n−m of ϵ n in Lemma 3.6.We observe that the operator ϵ m applies to C 1 , say The stages of the operation is given below.
The assertions (i) and (ii) follow.
4 Primitive factorization of maximal chains of (P (s) n , ) In this section we characterize the evacuation of the maximal chains of (P (s) n , ) in terms of a specific factorization of the chains.
For any integer ℓ ∈ [0, s] and a path B ∈ X n can be uniquely factorized into primitive blocks of certain widths as follows.Let We assume f 0 = 0.For 1 j b, let e j = f j − f j−1 .Then the path C can be factorized into primitive blocks

The evacuation of C ∈ X (s)
n can be determined by the evacuation of individual primitive blocks of C (Theorem 4.5).The following decomposition of ϵ n will be used in the proof of a characterization of the evacuation of primitive blocks (Proposition 4.3).Lemma 4.2.For any integer ℓ ∈ [0, s], the operator ϵ n can be decomposed as where

Proof. It suffices to consider the decomposition of the factor
We describe the process of the decomposition.
(i) From right to left, move the τ 1 of δ ℓ+2 to the right of the τ 2 of δ ℓ+3 and then move this τ 1 , along with the factor τ 1 τ 2 of δ ℓ+3 , to the right of the τ 3 of δ ℓ+4 .
The stages of the decomposition are given below.
p−1 denote the path obtained from C by moving the last step, which is a north step, all the way to the beginning.As a result, the remaining part of C is moved up one row and hence . The following result characterizes the evacuation of a primitive block, which leads to necessary conditions for a primitive block to be fixed by the operator ϵ n ; see Proposition 5.1.
Proof.Since C is a primitive ℓ-block, we observe that the segment C * goes from the point (0, s − ℓ) to the point (n − 1, p − ℓ) staying weakly above the line y = sx − ℓ, which is a maximal chain of the subposet of (P (s) n , ) induced on the set of points Making use of the decomposition of ϵ n in Lemma 4.2, we have the evacuation the electronic journal of combinatorics 25(1) (2018), #P1.33 The stages of the evacuation are described below.
(i) The initial s − ℓ steps are moved to the end by the operator (ii) Applying the operator (iii) By the operation of ρ 1 , the east step E in the nth column is moved up ℓ rows and hence d ′ n = s − ℓ.
(iv) By the operation of ρ 2 , the last step of C * ϵ n−1 , which is north, and ℓ − 1 north steps behind C * ϵ n−1 are moved to the front.As an equivalent result, the path C * ϵ n−1 is swapped with the ℓ north steps behind, and hence d ′ The proof is completed.
Theorem 4.5.For any path C ∈ X (s) be the primitive factorization of C for some integer b, where B j is a primitive ℓ j -block of width e j for 1 j b.Then the primitive factorization of Cϵ n is of the form Proof.By Proposition 4.3, B j ϵ e j is a primitive (s − ℓ j )-block for 1 j b.We prove the assertion by induction on the number of blocks of the primitive factorization.
The case b = 1 follows from Proposition 4.3.For b > 1, by Lemma 3.6 the operator ϵ n can be decomposed as ϵ n = ϵ e 1 γ n;e 1 ϵ n−e 1 .To find the evacuation of C, the operator ϵ e 1 applies to B 1 , leading to a primitive (s , where B ′ j = B j ϵ e j for 2 j b.The stages of operation are given below.
as required.

Enumeration of fixed points of the operator ϵ n
In this section we characterize and enumerate the maximal chains of P (s) n fixed by the operator ϵ n .First, we study the necessary conditions for a primitive block to be fixed under evacuation.(i) The integer s is even and ℓ = s 2 .
(ii) The integer n is odd and the path C passes the point 2 ) and the point

2
) and the point ( n−1 2 , sn 2 ), and so does C. (iii) For n = 1, the path D 1 D 2 is trivial.For n > 1 and the factorization , where p ′ = (s + 1)(n − 1), and hence the electronic journal of combinatorics 25(1) (2018), #P1.33 We describe the process as follows.The operator τ p ′ −1 n fixed by the operator ϵ n is uniquely determined by the segment from the origin through ( n−1 2 , s(n−1)
Corollary 5.2.Let D be a lattice path from the origin through ( n−1 2 , s(n−1) 2 n fixed by the operator ϵ n .
Proof.(ii) For n odd, if C = Cϵ n then the integer s is even and C passes the points ( n−1 2 , s(n−1)

2
) to the point some a ′ , where B j is a primitive ℓ j -block of width e j and ℓ 1 . Then C is the requested path in X
On the other hand, suppose D is a path from the origin through ( n−1 2 , s(n−1)

2
) to the point ( n−1 2 , sn 2 ).Then D has a unique factorization D = B 1 B 2 • • • B a ′ , for some a ′ , satisfying the following conditions.
• B j is a primitive ℓ j -block of width e j for 1 j a ′ − 1 and n fixed by the operator ϵ n .Now, we enumerate the fixed points of the map ϵ n : X the electronic journal of combinatorics 25(1) (2018), #P1.33

Concluding Remarks
Note that the poset (P (s) n , ) associated with the partition λ = (n s , (n − 1) s , . . ., 1) is the poset J(P ) constructed from the order ideals of a poset P .Stanley's result [8,Theorem 3.1] gives an alternative CSP for Fuss shapes of type A, which involves the q-polynomial W (q) in Eq. ( 2), the enumerator of linear extensions of P respecting the comajor index.For example, the poset associated with λ = (2, 1) ⊢ 3 can be constructed from the order ideals of the poset shown on the left hand side in Figure 8, with W (q) = 1+q + q 2 + q 3 + q 4 independent of the labeling of its elements (see Figure 9).However, the evacuation ϵ does not necessarily reverse the parity of the comajor index of linear extensions of P ; see Figure 9 (sometimes self-evacuating linear extensions have an odd comajor index).He proved that the evaluation W (−1) coincides with the number of domino linear extensions of P , i.e., the linear extensions ω ∈ L(P ) with the property ωτ p−1 τ p−3 τ p−5 • • • τ h = ω, where h = 1 if p is even, and h = 2 otherwise.To determine the self-evacuating linear extensions of P , he established a bijection ω → ω between the domino linear extensions ω and the self-evacuating linear extensions ω of P by where g = p − 1 if p is even, and g = p − 2 otherwise.For the poset P shown on the left hand side in Figure 8, the only domino linear extension ω corresponds to the maximal chain NNEE of J(P ) and the self-evacuating linear extension ωτ 1 •τ 3 τ 2 τ 1 corresponds to the maximal chain ENNE of J(P ).However, it is still unclear how to describe self-evacuating linear extension of a poset P explicitly.We contribute a neat characterization of the maximal chains of J(P ) fixed under evacuation for J(P ) = (P (s) n , ).As mentioned earlier, the map ϵ n does not necessarily reverse the parity of the statistic area of maximal chains of (P (s) n , ).This suggests the following problem.Problem 1. Find a statistic of s-Dyck paths (linear extensions of a poset, respectively) equidistributed with area (comaj, respectively) so that the evacuation is parity-reversing.
Among various cyclic sieving results on Catalan objects (e.g.[4, Theorem 7.1], [5, Theorem 8]), the case s = 1 in Theorem 2.1 gives an instance of CSP on a Catalan object, the triple (X n , X(q), ⟨ϵ n ⟩) of the poset associated with the partition λ = (n, n − 1, . . ., 1).Note that |X n | = c n+1 is the number of truncated Dyck paths of width n, where c n = the electronic journal of combinatorics 25(1) (2018), #P1.33 q q q q q q 2 q 2 q 2 q 2 q 2 q 3 q 3 q 3 q 3 q 3 q 4 q 4 q 4 q 4 q 4  is the nth Catalan number.It is worth mentioning that to our knowledge this result is the first instance using the area-enumerator X(q) as the q-polynomial while other known results using the q-analogue of Catalan number 1 [n+1]q 2n n q .By Theorem 5.4, for n even the paths C ∈ X n fixed by evacuation can be factorized as C = C 1 C 2 , where C 1 goes from the origin through ( n 2 , n 2 − 1) to ( n 2 , n 2 ).Moreover, Cϵ n = C ′ 2 C ′ 1 , where C ′ 1 = C 1 ϵ n 2 and C ′ 2 = C 2 ϵ n 2 .For example, inspecting the orbits of X 2 under evacuation shown in Figure 3, one can predict the two paths in X 4 fixed by evacuation, as shown in Figure 10.Recall that not all partitions λ of an integer n are good shapes, i.e., the triple (X , X(q), ⟨ϵ⟩) of the poset associated with λ exhibits CSP.Let g n be the number of the electronic journal of combinatorics 25(1) (2018), #P1.33 good shapes λ ⊢ n.We obtain the initial terms of the sequence {g n } n 0 by computer 1, 1, 2, 3, 5, 6, 11, 13, 21, 24, 40, 45, 71, 78, 122, 135, 202.A question might arise.Problem 2. Determine g n and characterize good shapes λ ⊢ n, with an explicit characterization of the fixed points under evacuation.
moved to the right by one column.(iii)Since d 1 s − ℓ, the first ℓ steps of C are north steps.First, we compute Cδ p−1 .Note that z 1 = N can be moved all the way to end subject to the conditiond j s − 1 for all j ∈ [2, n].As a result, the segment z 2 • • • z p of C is moved down one row and hence Π(Cδ p−1 ) = (d 1 + 1, . . ., d n + 1).Next, apply δ p−2 to the segment z 2 • • • z p ,leaving z 1 frozen in the last position.Continue in this way until z 1 , . . ., z ℓ are frozen in the back.The assertion follows.(iv) Since d n ℓ, the last ℓ steps of C are north steps.Note that δ * p−ℓ • • • δ * p−1 is the reverse operation of δ p−1 • • • δ p−ℓ and that the operator δ * p−ℓ moves the step z p−ℓ+1 = N all the way to the first position subject to no restriction on Π(C).As a result, the segment the electronic journal of combinatorics 25(1) (2018), #P1.33 z 1 • • • z p−ℓ of C is moved up one row.The assertion follows from the similar operations of δ * p−ℓ+1 , . . ., δ * p−1 .

Lemma 3 . 3 .
of paths with code (d 1 , . . ., d n ) such that d n ℓ, and let B n (ℓ) ⊆ X (s) n be the set of paths with code (d 1 , . . ., d n ) such that d j ℓ for all j ∈ [1, n].The following result is a property of the operator ϵ n , carrying paths with one restriction to paths with the other.The operator ϵ n establishes a bijection between A n (ℓ) and B n (s − ℓ).

Example 4 . 6 . 1 = B 1 ϵ 1 = 2 = B 2 ϵ 1 = 3 = B 3 ϵ 2 =
Let s = 3 and n = 4.Given the path C shown on the left hand side of Figure 5, let us construct the evacuation of C. As mentioned in Example 4.1, the primitive factorization of C consists of one primitive 3-block B 1 = ENNN and two primitive 2blocks B 2 = NENN and B 3 = NNNNEENN.Note that B ′ NNNE is a primitive 0-block and B ′ NNEN is a primitive 1-block.As shown in Example 4.4, B ′ NNNENNEN is a primitive 1-block.By Theorem 4.5, we have the primitive factorization of Cϵ 4 = B ′ 3 B ′ 2 B ′ 1 , shown on the right hand side of Figure 5.
) staying weakly above the line y = sx − s determines a unique path in X (s) n fixed by the operator ϵ n .Proof.Let C = B 1 • • • B b be the primitive factorization of C, where B j a primitive ℓ j -block of width e j for 1 j b.By Theorem 4.5, Cϵ n = B ′ b • • • B ′ 1 , where B ′ j is the evacuation of B j for 1 j b.If C = Cϵ n then ℓ j = s − ℓ b−j+1 and e j = e b−j+1 for 1 j b. (i) For n even, if b is odd, say b = 2a − 1, then B a = B ′ a = B a ϵ ea , which is fixed under evacuation.By Proposition 5.1, e a is odd.It follows that n = 2(e 1

n
fixed by the operator ϵ n .(ii) For n odd, if b is even, say b = 2a then n = 2(e 1 + • • • + e a ) is against the parity of n.Hence b is odd, say b = 2a − 1.Then B a = B a ϵ ea is fixed under evacuation.By Proposition 5.1, it follows that the path B a passes the points ( n−1 2 , s(n−1)

Figure 8 :
Figure 8: The paths in X 4 fixed by evacuation.

Figure 9 :
Figure 9: The comaj-enumerator W (q) of linear extensions of the poset shown on the left hand side of Figure 8, regarding labeling of its elements.

Figure 10 :
Figure 10: The paths in X 4 fixed by evacuation.
When γ n;m applies to C, by (i) and (iii) of Lemma 3.2, we observe that from z (s+1)m to z 1 the steps of C 1 are moved one by one to the back of C 2 by the operators ρ 1 , . . ., ρ (s+1)m accordingly and freeze in place afterwards, subject to the condition that the depth-code of C 2 meets the requirement in Lemma 3.2(iii) throughout the way.Notice that this is the case since Π(C 1 ) satisfies conditions (i) and (ii) and Π(C 2 ) satisfies condition (iii).
• • • τp′ 2 +1 moves the last step of D ′ 1 to the first position of D ′ 1 .Next, τp′ 2 applies to the two north steps in the middle of D ′ 2 (D ′ 1 ) ⊥ , leaving the path unchanged.Then the operator τp′ 2 −1 • • • τ 1 moves the last step of D ′ 2 to the first position of D ′ 2 .The assertion follows.
Then ℓ a = s − ℓ a+1 s − ℓ a and hence ℓ a 2 ).Then D has a unique primitive factorization, say with abuse of notation