$P$-strict promotion and $B$-bounded rowmotion, with applications to tableaux of many flavors

We define P-strict labelings for a finite poset P as a generalization of semistandard Young tableaux and show that promotion on these objects is in equivariant bijection with a toggle action on B-bounded Q-partitions of an associated poset Q. In many nice cases, this toggle action is conjugate to rowmotion. We apply this result to flagged tableaux, Gelfand-Tsetlin patterns, and symplectic tableaux, obtaining new cyclic sieving and homomesy conjectures. We also show P-strict promotion can be equivalently defined using Bender-Knuth and jeu de taquin perspectives.


Introduction
This paper builds on the papers [37,9,10] investigating ever more general domains in which promotion on tableaux (or tableaux-like objects) and rowmotion on order ideals (or generalizations of order ideals) correspond. In [37], N. Williams and the second author proved a general result about rowmotion and toggles which yielded an equivariant bijection between promotion on 2 × n standard Young tableaux and rowmotion on order ideals of the triangular poset n−1 (by reinterpreting the Type A case of a result of D. Armstrong, C. Stump, and H. Thomas [1] as a special case of a general theorem they showed about toggles). In [9], the second author, with K. Dilks and O. Pechenik, found a correspondence between a × b increasing tableaux with entries at most a + b + c − 1 under K-promotion and order ideals of [a] × [b] × [c] under rowmotion. In [10], the second and third authors with Dilks broadened this correspondence to generalized promotion on increasing labelings 1 arXiv:2012.12219v3 [math.CO] 4 Aug 2021 of any finite poset P with restriction function R on the labels and rowmotion on order ideals of a corresponding poset.
In this paper, we generalize from rowmotion on order ideals to rowmowtion on B-bounded Qpartitions and determine the corresponding promotion action on tableaux-like objects we call Pstrict labelings (named in analogy to column-strict tableaux). This general theorem includes all of the previously known correspondences between promotion and rowmotion and gives new corollaries relating P -strict promotion on flagged or symplectic tableaux to B-bounded rowmotion on nice Q-partitions. Our main results also specialize to include a result of A. Kirillov and A. Berenstein [24] which states that Bender-Knuth involutions on semistandard Young tableaux correspond to piecewise-linear toggles on the corresponding Gelfand-Tsetlin pattern.
The paper is structured as follows. The introduction begins in Section 1.1 with a motivating example. Then we define our new objects, P -strict labelings, and a corresponding promotion action in Section 1.2. In Section 1.3, we define B-bounded Q-partitions and the associated toggle and rowmotion actions. In Section 1.4, the final section of our introduction, we summarize the main results of this paper. Section 2 proves our main theorems relating P -strict promotion, toggles, and B-bounded rowmotion. Section 3 studies further properties of promotion and evacuation on P -strict labelings, including a jeu de taquin characterization of promotion for special P -strict labelings. Finally, Section 4 applies our main theorem to many special cases of interest.
1.1. An example. To motivate our main results, we begin with an example (see the remaining subsections of the introduction for definitions). In [10], Dilks and the second two authors found as an application of their main results an equivariant bijection between promotion on increasing labelings of a chain P = [n] := p 1 p 2 · · · p n with the label f (p j ) restricted as j ≤ f (p j ) ≤ 2j and rowmotion on order ideals of the positive root poset n . The idea for the current paper arose from the question of what happens in the above correspondence when order ideals of n are replaced by n -partitions of height (that is, weakly increasing labelings of n with labels in {0, 1, . . . , }).
In this paper, we give a bijection to the following: take copies of P = [n] to form the poset P × [ ] = {(p, i) | p ∈ P and 0 ≤ i ≤ } and consider labelings f : P × [ ] → N that are strictly increasing in each copy of P , weakly increasing along each copy of [ ], and obey the restriction j ≤ f (p j , i) ≤ 2j as before (call this restriction R). We call these P -strict labelings of P × [ ] with restriction function R. In this special case, under a mild transformation (represented by the top arrow of Figure 1), these are flagged tableaux of shape n with flag (2, 4, . . . , 2n) (that is, semistandard tableaux with entries in row j at most 2j). The rightmost arrow of Figure 1 represents the bijection from the first main result of this paper, Theorem 2.8.
Our second main result, Theorem 2.20, implies that P -strict promotion (also called flagged promotion in this case) on these flagged tableaux is in equivariant bijection with B-bounded rowmotion (also called piecewise-linear rowmotion) on these n -partitions with labels at most . Then we deduce by a theorem of D. Grinberg and T. Roby [19, Corollary 66] on birational rowmotion that promotion on these flagged tableaux is, surprisingly, of order 2(n + 1). Note there is no dependence on the number of columns ! We discuss this and other applications to flagged tableaux in more detail in Section 4.2. See Corollaries 4.28 and 4.30 for these specific results and Figure 1 for an example of the bijection.
1.2. Promotion on P -strict labelings. Promotion is a well-loved action defined by M.-P. Schützenberger on linear extensions of a finite poset [31]. On a partition-shaped poset, linear extensions are equivalent to standard Young tableaux. Promotion has been defined on many other flavors of tableaux and labelings of posets using jeu de taquin slides and their generalizations. Equivalently (as shown in [17,34,9,10]), promotion may be defined by a sequence of involutions, introduced by E. Bender and D. Knuth on semistandard Young tableaux [2]. This will be our main perspective; we discuss the jeu de taquin viewpoint further in Section 3. Below, we define P-strict labelings, which generalize both semistandard Young tableaux and increasing labelings. We extend the definition of promotion in terms of Bender-Knuth involutions to this setting. We show in Theorem 3.10 in which cases promotion may be equivalently defined using jeu de taquin. Definition 1.1. In this paper, P represents a finite poset with partial order ≤ P , indicates a covering relation in a poset, and q are positive integers, [ ] denotes a chain poset (total order) of elements (whose elements will be named as indicated in context), and P × [ ] = {(p, i) | p ∈ P, i ∈ N, and 1 ≤ i ≤ } with the usual Cartesian product poset structure.
Below, we define P -strict labelings on convex subposets of P × [ ]. A convex subposet is a subposet such that if two comparable poset elements a and b are in the subposet, then so is the entire interval [a, b]. This level of generality is necessary to, for instance, capture the case of promotion on semistandard Young tableaux of non-rectangular shape. Proof. Since S is convex, along any fiber F p we have (p, i) ∈ S with i 0 < i < i 1 for some i 0 ≥ 0 and some i 1 ≤ + 1. If F p = ∅, let u(p) = i 0 and v(p) = + 1 − i 1 . If ω P p, then u(ω) ≤ u(p), otherwise (p, u(ω)), (ω, u(ω) + 1) ∈ S but (ω, u(ω)) / ∈ S, contradicting the convexity of S. Similarly, v(ω) ≥ v(p). If F p = ∅, then F ω = ∅ for all ω P p by convexity. For all p ∈ P with F p = ∅, set u(p) = min{u(q) | F q = ∅} and v(p) = − u(p). Thus u(p) + v(p) = and, over all of P , u(p 1 ) ≥ u(p 2 ) when p 1 P p 2 . Moreover, since for all p with F p = ∅ we have v(p) < − u(p), v(p 1 ) ≤ v(p 2 ) for all p 1 P p 2 .
In this paper, R will always represent a restriction function. Definition 1.7. We say that a function f : That is, f is strictly increasing inside each copy of P (layer), weakly increasing along each copy of the chain [ ] (fiber), and such that the labels come from the restriction function R. Let The following definition says that R is consistent if every possible label is used in some P -strict labeling.
Definition 1.8. Let R : P → P(Z). We say R is consistent with respect to P × [ ] v u if, for every p ∈ P and k ∈ R(p), there exists some P -strict labeling f ∈ L P ×[ ] (u, v, R) and u(p) < i < +1−v(p) such that f (p, i) = k.
We denote the consistent restriction function induced by (either global or local) upper and lower bounds as R b a , where a, b : P → Z. In the case of a global upper bound q, our restriction function will be R q 1 , that is, we take a to be the constant function 1 and b to be the constant function q. Since a lower bound of 1 is used frequently, we suppress the subscript 1; that is, if no subscript appears, we take it to be 1. [10]. A notion of consistent R for this case was defined. This coincides with the above definition.
We will use the following two definitions in Definition 1.12. Definition 1.10. Let R(p) >k denote the smallest label of R(p) that is larger than k, and let R(p) <k denote the largest label of R(p) less than k.
It is important to note that the above definition is analogous to the increasing labeling case of [10], so raisability (lowerability) is thought of with respect to the layer, not the entire P -strict labeling. Definition 1.12. Let the action of the kth Bender-Knuth involution ρ k on a P -strict labeling f ∈ L P ×[ ] (u, v, R) be as follows: identify all raisable labels f (p, i) = k and all lowerable labels f (p, i) = R(p) >k (if k = max R(p), then there are no raisable or lowerable labels on the fiber F p ). Call these labels 'free'. Suppose the labels f (F p ) include a free k labels followed by b free R(p) >k labels; ρ k changes these labels to b copies of k followed by a copies of R(p) >k . Promotion on Pstrict labelings is defined as the composition of these involutions: Pro(f ) = · · · • ρ 3 • ρ 2 • ρ 1 • · · · (f ). Note that since R induces upper and lower bounds on the labels, only a finite number of Bender-Knuth involutions act nontrivially.
We compute promotion on a P -strict labeling in Figure 2. We continue this example in Figure 5.
Example 1.13. Consider the action of ρ 1 in Figure 2. In the fiber F a , neither of the 1 labels can be raised to R(a) >1 = 3, since they are restricted above by the 3 labels in the fiber F b . However, the 3 label in F a can be lowered to a 1, and so the action of ρ 1 takes the one free 3 label and replaces it with a 1. Similarly, in F c , the 2 is lowered to a 1. In F b , the 1 can be raised to a 3 and the 3 can be lowered to a 1. Because there is one of each, ρ 1 makes no change in F b .
After applying ρ 2 , we look closer at the action of ρ 3 . In F a , there are no 3 labels or R(a) >3 = 4 labels, so we do nothing. In F b , however, there are three 3 labels that can be raised to R(b) >3 = 5 and one 5 that can be lowered to 3. Thus ρ 3 replaces these four free labels with one 3 and three 5 labels.
Remark 1.14. In the case = 1, L P ×[ ] (R) equals Inc R (P ), the set of increasing labelings of P with restriction function R. So the above definition specializes to generalized Bender-Knuth involutions and increasing labeling promotion IncPro, as studied in [10]. If, in addition, P is (skew-)partition shaped, these increasing labelings are equivalent to (skew-)increasing tableaux, and the above definition specializes to K-Bender-Knuth involutions and K-Promotion, as in [9].
If we restrict our attention to linear extensions of P , the above definition specializes to usual Bender-Knuth involutions and promotion, as studied in [34].
If P = [n] and is arbitrary, L P ×[ ] (R q ) is equivalent to the set of semistandard Young tableaux of shape an n× rectangle and entries at most q, and L P ×[ ] (u, v, R q ) is the set of (skew-)semistandard Young tableaux with shape corresponding to P × [ ] v u and entries at most q. In these cases, the above definition specializes to usual Bender-Knuth involutions and promotion. We give more details on this specialization in Section 4.1.
Given that Definition 1.12 specializes to the right thing in each of these cases (including linear extensions and semistandard Young tableaux), we will no longer use the prefixes K-, increasing labeling, or generalized, and rather call all these actions 'Bender-Knuth involutions' and 'promotion', letting the object acted upon specify the context.

Rowmotion on Q-partitions.
Rowmotion is an intriguing action that has recently generated significant interest as a prototypical action in dynamical algebraic combinatorics; see, for example, the survey articles [29,36]. Rowmotion was originally defined on hypergraphs by P. Duchet [11] and generalized to order ideals J(Q) of an arbitrary finite poset (Q, ≤ Q ) by A. Brouwer and A. Schrijver [4]. P. Cameron and D. Fon-der-Flaass [5] then described it in terms of toggles; thereafter, Williams and the second author [37] related it to promotion and gave it the name 'rowmotion'. Rowmotion was further generalized to piecewise-linear and birational domains by D. Einstein and J. Propp [12,13]. In this paper, we discuss toggling and rowmotion on Q-partitions, as a rescaling of the piecewise-linear version.
In light of our use of P for P -strict labelings, we use Q rather than P when referring to an arbitrary finite poset associated with the definitions of this section. Remark 1.21. Let Q be the poset Q with two additional elements added for each x ∈ Q: a minimal element0 x covered by x and a maximal element1 Remark 1.22. Note that B-bounded Q-partitions correspond to rational points in a certain marked order polytope, though this perspective is not necessary for this paper.
In Definitions 1.23 and 1.25 below, we define toggles and rowmotion. In the case of A (Q), these definitions are equivalent (by rescaling) to those first given by Einstein and Propp on the order polytope [12,13]. By the above remarks, it is sufficient to give the definitions of toggles and rowmotion for A B (Q).
Remark 1.24. By the same reasoning as in the case of order ideal toggles, the τ x satisfy: (1) τ 2 x = 1, and (2) τ x and τ x commute whenever x and x do not share a covering relation.
Remark 1.26. It may be argued that we should call these actions piecewise-linear toggles and piecewise-linear rowmotion as defined in [12,13], but as in the case of promotion on tableaux and labelings, unless clarification is needed, we choose to leave the names of these actions adjective-free, allowing the objects acted upon to indicate the context. 1.4. Summary of main results. Our first main theorem gives a correspondence between Pstrict labelings L P ×[ ] (u, v, R) under promotion and specificB-bounded Q-partitions A B (Q) under a composition of toggles, namely, the toggle-promotion TogPro of Definition 2.6. Here Q is the poset Γ(P,R) constructed in Section 2.1 andB depends on u, v, and R. The bijection map Φ is given in Definition 2.9. See Figure 5 for an illustration of this theorem and Figure 6 for an example of Φ.
Our second main theorem specifies cases in which toggle-promotion is conjugate in the toggle group to rowmotion, namely, when A B (Γ(P,R)) is column-adjacent (see Definition 2.19).
Column-adjacency holds in many cases of interest, including the case of restriction functions induced by global or local bounds, such as the various sets of tableaux discussed in Section 4.
Our third main theorem states that in the case of a global upper bound q, P -strict promotion can be equivalently defined in terms of jeu de taquin; see Definition 3.1 and Figure 7. In this same special case, we define and study P -strict evacuation; see Section 3.2. We highlight some corollaries of our main theorems. The first is a correspondence that has been noted before (see Remark 4.16) between promotion on rectangular semistandard Young tableaux SSYT( n , q) and rowmotion on Q-partitions A (Q), where Q is a product of two chains poset (see Figure 12). Such correspondences are often of interest since they provide immediate translation of results, such as Rhoades' cyclic sieving theorem on SSYT( n , q) [28], from one domain to the other. We also recover the following result of Kirillov and Berenstein relating Bender-Knuth involutions ρ k on semistandard Young tableaux SSYT(λ/µ, q) with elementary transformations t k on Gelfand-Tsetlin patterns GT(λ,μ, q) (see Figure 11). . The set SSYT(λ/µ, q) is in bijection with GT(λ,μ, q), wherẽ λ i := λ 1 − µ n−i+1 andμ i := λ 1 − λ n−i+1 . Moreover, ρ k on SSYT(λ/µ, q) corresponds to t q−k on GT(λ,μ, q).
Theorem 2.20 specializes to the following two corollaries on flagged tableaux FT(λ, b) of shape λ and flag b. The first was our motivating example of Subsection 1.1 and Figure 1; the second involves flagged tableaux of staircase shape that have appeared in the literature [7,32].  The first corollary enables us to translate an existing cyclic sieving conjecture on A ( n ) [23] to these flagged tableaux (see Conjecture 4.32). The second allows us to translate an existing cyclic sieving conjecture on flagged tableaux [7,32] to (δ, )-bounded Q-partitions A δ (Q), where Q is a product of two chains poset (see Conjecture 4.42 and Figure 13). These translations provide new perspectives on the conjectures, which may be helpful for proving them.
We also present a new conjecture regarding homomesy on A ( n ) and use our main theorem to translate it to flagged tableaux in Conjecture 4.37.
Conjecture 4.35. The triple A ( n ), TogPro, R is 0-mesic when n is even and 2 -mesic when n is odd, where R is the rank-alternating label sum statistic.
Finally, we obtain the following correspondence between promotion on symplectic tableaux of staircase shape sc n and rowmotion on (δ, )-bounded Q-partitions A δ (Q), where Q is the triangular poset n (see Figure 14). Corollary 4.50. There is an equivariant bijection between Sp(sc n , 2n) under Pro and A δ ( n ) under Row, where for (i, j) ∈ n , δ(i, j) = min(j, n) and (i, j) = i − 1.
This correspondence shows the cardinality of A δ ( n ) is 2 n 2 , as a consequence of the symplectic hook-content formula of P. Campbell and A. Stokke [6].

P -strict promotion and rowmotion
In this section, we prove our first two main theorems. Theorem 2.8 relates promotion on P -strict labelings with restriction function R and toggle-promotion on B-bounded Q-partitions, where Q is the poset Γ(P,R), whose construction we discuss in the next subsection. Theorem 2.20 extends this correspondence to rowmotion in the case when our poset is column-adjacent.    . Let P be a poset and R : P → P(Z) a (not necessarily consistent) map of possible labels. Then define Γ(P, R) to be the poset whose elements are (p, k) with p ∈ P and k ∈ R(p) * , and covering relations given by (p 1 , k 1 ) (p 2 , k 2 ) if and only if either (1) p 1 = p 2 and R(p 1 ) >k 2 = k 1 (i.e., k 1 is the next largest possible label after k 2 ), or (2) p 1 p 2 (in P ), k 1 = R(p 1 ) <k 2 = max(R(p 1 )), and no greater k in R(p 2 ) has k 1 = R(p 1 ) <k .
That is to say, k 1 is the largest label of R(p 1 ) less than k 2 (k 1 = max(R(p 1 ))), and there is no greater k ∈ R(p 2 ) having k 1 as the largest label of R(p 1 ) less than k.
Example 2.3. Refer to Figure 3. The poset Γ(P, R) consists of four chains corresponding to each element a, b, c, and d, where each chain contains one less element than R(p). For instance, R(a) = {1, 3, 4}, so, by (1) in Definition 2.2, Γ(P, R) contains the chain (a, 3) (a, 1). There is no element (a, 4) since 4 = max R(a) and is therefore not in R(a) * . We indicate this omission by writing a, 4 beneath the element (a, 3). The covering relations between the elements in these chains are described by (2) in Definition 2.2. For example, (b, 1) (d, 2) since b d and 1 is the greatest element of R(b) that is strictly less than 2. Note (d, 6) does not cover (b, 3) since 5 ∈ R(b) is the greatest element less than 6, not 3.
In [10,Theorem 4.31] it is shown that if R consistent on P , increasing labelings on P under increasing labeling promotion are in equivariant bijection with order ideals of Γ(P, R) under togglepromotion. This correspondence drives our first main theorem. In order to apply this result from [10] to P -strict labelings, we need a restriction function that is consistent on P , not just on P × [ ] v u . The next definition constructs such a restriction function.
u . Denote the number of elements less than or equal to p in a maximum length chain containing p as h(p) and the number of elements greater than or equal to p in a maximum length chain containing p ash(p). Define a new restriction functionR on P given bŷ u , thenR is consistent on P .
Proof. If p 1 < P p 2 , then min q∈P R(q) −h(p 1 ), an element ofR(p 1 ), is less than all elements of R(p 2 ) and max q∈P R(q) + h(p 2 ), an element ofR(p 2 ), is greater than all elements ofR(p 1 ). Thus, for any p ∈ P and any element k ofR(p ), the labeling f of P given by is an element of L P × [1] (R) = Inc R (P ) (see Remark 1.9). Since for all p ∈ P and k ∈R(p) there exists a labeling f with f (p) = k,R is consistent on P .
We use the structure of Γ(P,R) in our main result. While any consistent restriction function on P constructed by adding a new minimum and maximum element to each R(p) would serve our purposes, we choose to useR for the sake of consistency.

2.2.
First main theorem: P -strict promotion and toggle-promotion. Below, we state and prove our first main result, Theorem 2.8. First, we define an action on B-bounded Γ(P,R)partitions.
This composition is well-defined, since the toggles within each τ k commute by Remark 1.24.
To see an example of toggle-promotion on aB-bounded Γ(P,R)-partition, refer to Figure 4. See Figure 5 for an example illustrating Theorem 2.8.
In Theorem 2.8 below, Φ is the bijection map given in Definition 2.9.
Theorem 2.8. The set of P -strict labelings The proof will use the following definitions and lemmas. We first define the bijection map.
Definition 2.9. We define the map Φ :       Figure 4 for the steps in calculating TogPro in this example.
The map φ 2 in Definition 2.9 is the main bijection used in [10, Theorem 2.14], and the map φ 3 is the usual bijection between multichains of J(P ) and P -partitions (see [33]).
Lemma 2.10. The map Φ is well-defined and invertible.
For invertibility, φ 1 is invertible by removing the labels off that are not in R, and φ 2 is invertible by [10]. Given σ ∈ A B (Γ(P,R)) we can recover the associated multichain by From the definition of φ 1 , the number of minR(p) labels in f (F p ), or the number of order ideals in φ 2 (f ) containing (p, minR(p)), is u(p). Therefore, Φ(f )(p, minR(p)) = − u(p). Next, (p, max R(p) * ) is included in every order ideal associated to a layer where p is not labeled by maxR(p). Since there are v(p) such layers, v(p) order ideals do not contain (p, max R(p) * ), so Φ(f )(p, maxR(p) * ) = v(p).
Lemma 2.11. The bijection map Φ equivariantly takes the generalized Bender-Knuth involution ρ k to the toggle operator τ k .
The following notation will be useful for the proof of this lemma. Definition 2.12. We consider the label at (p, i) ∈ P × [ ] v u to be in position i, and the first (last) position satisfying a particular condition is the least (greatest) such position.
That is, j p k is the first position in the fiber F p with label greater than k, where we may consider a label at (p, − v(p) + 1) that is greater than all other labels.
Example 2.14. In Figure 6, we have j c 4 = 3, j a 4 = 5, and j b −1 = 2. We can now write the bijection Φ in terms of j p k .
Proof of Lemma 2.11. We prove this lemma by showing Φ equivariantly takes the action of ρ k on f (F p ) to the toggle τ k at (p, k) ∈ Γ(P,R). Let f ∈ L P ×[ ] (u, v, R) and Φ(f ) = σ ∈ A B (Γ(P,R)). Consider the action of ρ k on f (F p ). If k / ∈ R(p) * , then ρ k acts as the identity on f (F p ) and τ k acts as the identity on σ(p, k), so we are done. Therefore, let k ∈ R(p) * . We aim to count the number of raisable k labels and lowerableR(p) >k labels in f (F p ). We begin with finding the number of raisable k.
Using Lemma 2.15, the total number of (not necessarily raisable) k labels is given by Now, we determine which of these labels are raisable. We consider three cases based on the upper covers of (p, k) in Γ(P,R) associated to a different element of P .
Case U = ∅: For (ω, c) ∈ U, by construction of Γ(P,R), k =R(p) <c and c is the largest such c ∈R(ω). Equivalently, c is the greatest element ofR(ω) that is less than or equal toR(p) >k . Thus, sinceR(ω) >c >R(p) >k , the first position in f (F p ) that is not restricted above by labels in f (F ω ) is j ω c . Therefore the first position in f (F ω ) that can be raised toR(p) >k is max (ω,c)∈U j ω c , so the number of labels in f (F p ) that can be raised toR(p) >k (that are necessarily less thanR(p) >k ) is Case U = ∅ and ω P p for some ω ∈ P : This implies that k =R(p) <c for any c ∈R(ω) for any ω P p. Thus, if c > k, then we also have c >R(p) >k . Since f is strict on layers, if f (p, i) = k, all f (ω, i) are greater than R(p) >k . Therefore all k labels in f (F p ) are raisable.
Case U = ∅ and p has no upper covers in P : In this case, f (F p ) is not restricted above, and again all k labels in f (F p ) are raisable.
The number of raisable k is the lesser of the number of k labels and the number of labels less thanR(p) >k that can be raised toR(p) >k . Let Y = {y | y covers (p, k) in Γ(P,R)}. Then, by the above cases, the number of raisable k in f (F p ) is given by Suppose there are a raisable k and b lowerableR(p) >k in f (F p ). Apply ρ k to f (F p ), and let σ p k be the Γ(P,R)-partition corresponding to this new P -strict labeling. For all d = k, the first position in f (F p ) with a label greater than d is unchanged after applying ρ k . Thus the only label that differs between σ and σ p k is the label at (p, k). Since there are b raisable k in ρ k (f (F p )), with Y and Z defined as above we have which is exactly τ (p,k) (σ)(p, k). Thus, ρ k on f corresponds to toggling on σ over all elements (p, k) with p ∈ P .
In the following, we give an example of each case from the previous proof. Example 2.17. Figure 6 shows an example of the bijection map; the number of 3 labels in We now prove our first main theorem.
Proof of Theorem 2.8. By Lemma 2.10, Φ is a bijection. By 2.3. Second main theorem: P -strict promotion and rowmotion. Our next main result, Theorem 2.20, says that for certain kinds of restriction functions, promotion on P -strict labelings of Definition 2.19. We say that A B (Γ(P, R)) is column-adjacent if whenever (p 1 , k 1 ) (p 2 , k 2 ) in Γ(P, R) and neither of (p 1 , k 1 ) nor (p 2 , k 2 ) are fixed in A B (Γ(P, R)), then |k 2 − k 1 | = 1.
We call this column-adjacent because it implies that the non-fixed poset elements (p, k) of Γ(P, R) can be partitioned into subsets indexed by k, called columns, whose elements have covering relations with other non-fixed elements only when they are in adjacent columns. For many nice cases, including the posets considered in Section 4, the word column is visually appropriate.
To any σ ∈ A B (Γ(P,R)) we associate aΓ(P,R)-partitionσ in A (Γ(P,R)) whereσ(p, k) = σ(p, k). We define the toggleτ (p,k) on A (Γ(P,R)) as usual with the added restriction that, if (p , k ) (p, k) in Γ(P,R) and (p , k ) is fixed in A B (Γ(P,R)) with σ(p , k ) = a for all σ, then the minimum value of the upper covers of (p, k) may not exceed a, and, similarly, if (p , k ) (p, k) in Γ(P,R) and (p , k ) is fixed as a in A B (Γ(P,R)), the maximum value of the lower covers must be at least a. Thus,τ (p,k) (σ)(p, k) = τ (p,k) (σ)(p, k).
Since these toggles onΓ(P,R)-partitions share the same commutation relations as toggles on J(Γ(P,R)), as noted in Remark 1.24, we can apply a conjugation result from [10] as follows. Because A B (Γ(P,R)) is column-adjacent, if (p, k) (p , k ) inΓ(P,R), then |k − k| = 1. Now, ifτ k is the composition over all p ∈ P ofτ (p,k) , TogPro = · · · •τ 1 •τ 0 •τ −1 • · · · is conjugate to Row oñ Γ(P,R) by [10,Theorem 4.19]. Therefore, TogPro is also conjugate to Row on A B (Γ(P,R)), and we obtain the result by Theorem 2.8. Remark 2.21. As long as a toggle order is a column toggle order, as defined in [10], the composition of toggles will be equivariant with rowmotion, so there are many more toggle orders besides that of TogPro that are conjugate to rowmotion. We do not need this full level of generality of toggle orders.
We show in the following proposition that for the case where our restriction function is induced by upper and lower bounds for each element (this includes the case of a global bound q), we have the column-adjacent property, so Theorem 2.20 yields Corollary 2.24.
is column-adjacent. The proof of the above uses the following lemma.
If k + 1 and k + 2 ∈ R b a (p ), then, because P × [ ] v u is a convex subposet, any position in the fiber f (F p ) that can be labeled by k + 1 can also be labeled by k + 2. Thus, if k + 1 and . Therefore, there must exist p 1 of the covers p such that k + 2 / ∈ R(p 1 ). Moreover, if σ(p, k) < σ(p 1 , k + 1), by Lemma 2.15, the first position greater than k + 1 in f (F p 1 ) occurs before the first position greater than k in f (F p ). In this position, any values greater than k + 1 and less than R b a (p 1 ) ≥k+1 , including k + 2, would be possible, a contradiction. Thus σ(p, k) = σ(p 1 , k + 1). Now, either (p 1 , k + 1) is fixed byB and we are done, or, by the above reasoning, there exists p 2 P p 1 such that σ(p 1 , k + 1) = σ(p 2 , k + 2) and k + 3 / ∈ R b a (p 2 ). We continue this until there exists a maximal p m ∈ P such that σ(p, k) = σ(p 1 , k + 1) = · · · = σ(p m , k + m) and k + m is fixed byB. Therefore σ(p, k) is always equal to the value of a fixed element, and since σ was arbitrary, (p, k) is fixed in A B (Γ(P, R b a )).
Proof of Proposition 2. 22. We show that if (p 1 , k 1 ) (p 2 , k 2 ) in Γ(P, R b a ) and |k 2 − k 1 | > 1, then either (p 1 , k 1 ) or (p 2 , k 2 ) is fixed. Without loss of generality, let k 2 − k 1 > 1. If p 1 = p 2 , then . Thus, by Lemma 2.23 again, (p 1 , k 1 ) is fixed.  [10]. However, even though R b a is induced by lower and upper bounds, this is not always the case. The requirement that R b a be consistent on P × [ ] v u can result in gaps in a particular R b a (p) depending on u and v. As an example, consider the semistandard Young tableau with shape (4, 4, 4, 4, 2, 2, 2)/(2, 2, 2) and global maximum 5 (that is, P = [7] with restriction function R 5 1 and u, v determined by the shape). In this case, the fourth row of the tableau can only be labeled by elements of {1, 2, 4, 5}.
2.4. Special cases of A B (Γ(P,R)). In this subsection, we consider cases in which A B (Γ(P,R)) from our main theorem can be more nicely described by restricting certain parameters. We begin with two propositions that show when A B (Γ(P,R)) is equivalent to A (Γ(P, R)) or A δ (Γ(P, R)) from Definitions 1.15 and 1.17, and conclude with a corollary of our main theorem in the case where A B (Γ(P,R)) is simply the product of the poset P with a chain. We use these results several times in Section 4.
Proof. We first consider the covering relations of the elements (p,k) in Γ(P,R) given by condition (2) in Therefore the only covering relations in Γ(P,R) between elements of Γ(P,R) \ Γ(P, R) and Γ(P, R) are given by (1) in Definition 2.2. Specifically, these are (p, min R(p)) (p, minR(p) * ) and (p, maxR(p) * ) (p, max R(p)) for all p ∈ P .
Lemma 2.28. Let P be a graded poset of rank n. Then Γ(P, R q ) is isomorphic to P × [q − n − 1] as a poset. j ) if and only if p = p and j = j − 1 or j = j and p p (that is, the usual ordering (p, j) ≤ (p , j ) if and only if p ≤ P p and j ≤ j ).

P -strict promotion and evacuation
In this section, we define promotion on P -strict labelings L P ×[ ] (u, v, R q ) via jeu de taquin and prove Theorem 3.10, which shows this is equivalent to our promotion via Bender-Knuth involutions from Definition 1.12. We also define evacuation on P -strict labelings and show some properties of evacuation in this setting.
3.1. Third main theorem: P -strict promotion via jeu de taquin. We begin with the definition of jeu de taquin promotion on P -strict labelings L P ×[ ] (u, v, R q ).
i g(p, k) = and g(p , k) = i for some p P p (1a) i g(p, k) = , g(p, k + 1) = i, and g(p , k + 1) = for any p P p (1b) g(p, k) = i and g(p , k) = for some p P p (1c) g(p, k) = i, g(p, k − 1) = , and g(p , k − 1) = i for any p P p (1d) g(p, k) otherwise. (1e) In words, jdt i (g) replaces a label at (p, k) with i if i is the label of a cover of (p, k) in its layer, or if i is the label of a cover of (p, k) in its fiber and this cover does not also cover an element within its own layer labeled by . Furthermore, jdt i (g) replaces a label i by if (p, k) covers an element in its layer labeled by , or replaces a label i by if (p, k) covers an element in its fiber labeled by , provided said element is not covered by an element in its layer labeled with i. Aside from these cases, jdt i (g) leaves all other labels unchanged.
Let jdt i→j : Z (P ) → Z (P ) be defined as In words, jdt i→j (g)(x) replaces all labels i by j. For That is, first replace all 1 labels with . Then perform the ith jeu de taquin slide jdt i times for each 2 ≤ i ≤ q. Next, replace all labels with q + 1. Define jeu de taquin promotion on f as JdtPro(f )(x) = jdt(f )(x) − 1.
In Proposition 3.5, we show that if we begin with a P -strict labeling f , JdtPro(f ) is always a P -strict labeling. In order to prove this, we need Lemmas 3.3 and 3.4, which give us conditions that a labeling cannot violate when performing jeu de taquin slides.
When performing a jeu de taquin slide of JdtPro(f ), no integer labels can violate the P -strict labeling order relations.
Proof. Because we apply all jeu de taquin slides jdt 2 , then all jeu de taquin slides jdt 3 , and so on for each jdt i where 2 ≤ i ≤ q, each time is replaced by a number, that number is the smallest label of its covers. As a result, no integer labels can violate the order relations after performing a jeu de taquin slide.
is obtained by performing jeu de taquin slides on f , we can never have jdt i (g)(p, k) = jdt i (g)(p , k) = when p > P p.
Proof. We show the claim by contradiction. Suppose jdt i (g)(p, k) = jdt i (g)(p , k) = for some p > P p. Furthermore, assume this is the first application of a jeu de taquin slide for which this occurs. In other words, we do not have two comparable elements within the same layer that both have a label of prior to this application of jdt i . Suppose this occurs from (1c) of Definition 3.1. This implies g(p , k) = for some p P p, which cannot occur by our assumption that jdt i (g)(p, k) = jdt i (g)(p , k) = is the first application of a jeu de taquin slide for which we have comparable elements within the same layer that are both labeled with . Now assume jdt i (g)(p, k) = jdt i (g)(p , k) = occurs after applying (1d) of Definition 3.1. For this to occur, we would need either g(p, k) = i and g(p , k) = , or g(p, k) = and g(p , k) = i. However, by assumption, any element between (p, k) and (p , k) cannot be labeled with . Furthermore, by Lemma 3.3, we cannot have any integer labels violate the order relations, so any element between (p, k) and (p , k) cannot be labeled with i. As a result, we may assume p P p. We can eliminate g(p, k) = and g(p , k) = i as a possibility, as (1a) of Definition 3.1 would be applied to g(p, k), resulting in jdt i (g)(p, k) = i. Therefore, we may assume g(p, k) = i and g(p , k) = . We may also assume g(p, k − 1) = in order for (1d) of Definition 3.1 to be invoked. However, by our assumption, this means g(p , k − 1) cannot have label , implying that g(p , k − 1) = i. By definition, (1d) of Definition 3.1 cannot be applied. We obtained a contradiction with each of (1c) and (1d) of Definition 3.1, implying that we cannot have jdt i (g)(p, k) = jdt i (g)(p , k) = for some p > P p.

Proof. By construction, JdtPro(f ) is a labeling of P × [ ] v
u with integers in {1, . . . , q}. By the definition of JdtPro(f ), we perform each jeu de taquin slide times. Note that we only need to perform each jdt i until the labels are above the i labels in every fiber where both appear. This is guaranteed to happen if we perform it times, as every fiber is of length at most . We only need to verify that JdtPro(f ) has the order relations of a P -strict labeling. By Lemma 3.3, no integer labels of JdtPro(f ) can violate the order relations after performing a jeu de taquin slide. Additionally, by Lemma 3.4, if g ∈ Z (P × [ ] v u ) is obtained by performing jeu de taquin slides on f , we can never have jdt i (g)(p, k) = jdt i (g)(p , k) = when p > P p. Because of this, we guarantee that no q + 1 labels violate the order relations after performing jdt →(q+1) as part of JdtPro. As a result, this means the strict order relations of the P -strict labeling will be satisfied when we perform jdt →(q+1) .
Our goal is Theorem 3.10, which states that jeu de taquin promotion from Definition 3.1 coincides with our definition of promotion by Bender-Knuth involutions. The crux of the proof is Lemmas 3.6 and 3.8. The idea of Lemma 3.6 is as follows. By definition, when performing JdtPro(f ), we perform each jeu de taquin slide times. We observe that for f ∈ L P ×[ ] (u, v, R q ), when we apply jdt i , cases (1a) and (1c) of Definition 3.1 can only be invoked on the first application of jdt i . Proof. We begin by proving the result for jdt q . Suppose g(p, k) = . If there is a cover (p , k) of (p, k) in the kth layer of P × [ ] v u , then we must have g(p , k) = q, as g(p , k) could not be less than q nor could it be by Lemma 3.4. Furthermore, if there does not exist a cover (p , k) of (p, k) in the kth layer, neither (1a) nor (1c) is invoked on from g(p, k) when applying jdt q . Therefore, we may assume a cover of (p, k) in the kth layer has a label of q. In other words, we assume there exists a p p such that g(p , k) = q. When applying jdt q , the first application of jdt q will invoke (1a) and (1c), resulting in g(p , k) being labeled with for any labels g(p , k) such that p p and g(p , k) = q. However, on subsequent applications of jdt q , (1a) cannot be invoked to result in a for any g(p , k) where p p . This is because g(p , k), a label for a cover of (p , k) in the kth layer, would need to be labeled with either q or , neither of which are possible due to Lemma 3.4. This means there does not exist a cover (p , k) of (p , k) in the kth layer at all, as g(p , k) also cannot be less than q.
We might be concerned that subsequent invocations of (1b) or (1d) within the fiber F p results in a appearing in a layer with which (1c) can be invoked for a second time. However, because there is no (p , k) ∈ P × [ ] v u , there cannot be an element (p , k ) ∈ P × [ ] v u in any layer k where k > k by definition of v. Hence, subsequent invocations of (1b) or (1d) cannot position a into a separate layer such that (1c) can be invoked for a second time. As a result, for this case, the label of g(p, k) can affect the label of a separate fiber only on the first application of jdt q via (1a) and (1c). An analogous argument shows that if we begin with g(p, k) = q, the label of q can only affect the label of a separate fiber on the first application of jdt q .
We have shown that when applying jdt q in JdtPro(f ), (1a) and (1c) of Definition 3.1 can only be invoked on the first application of jdt q . To show the result for any jdt i , let f ≤i with restriction R i denote the P -strict labeling f restricted to the subposet of elements with labels less than or equal to i. Because f ≤i has restriction function R i , (1a) and (1c) of Definition 3.1 can only be invoked on the first application of jdt i in JdtPro(f ≤i ), which means these cases can only be invoked on the first application of jdt i in JdtPro(f ).
In order to state Lemma 3.8, we need the following definition.
then reduce all unfrozen labels by 1. We clarify that boxes labeled i + 1 from the step jdt →(i+1) are considered unfrozen.
To prove Theorem 3.10, it will be sufficient to show that applying JdtPro q−1 and the Bender-Knuth involution ρ q yields the same result as JdtPro itself.
First, consider the case that there are no boxes in f . This implies that there were no elements labeled 1 in f , so JdtPro(f ) reduces all labels by 1. On the other hand, JdtPro q−1 (f ) will reduce all labels by 1 except labels that are q, as these labels are frozen. However, after reducing unfrozen labels, there are no elements with a label of q − 1, which means ρ q−1 changes all labels of q to q − 1. The cumulative effect is that all labels in f are reduced by 1. Therefore, in this case, we have JdtPro(f ) = ρ q−1 • JdtPro q−1 (f ).
We now consider the case where f has at least one element labeled . When applying jdt q , a label can only change if it is or q. By Lemma 3.6, when applying jdt q , (1a) and (1c) of Definition 3.1 can only be invoked on the first application of jdt q . We now show that when applying jdt q , the first application of jdt q places the correct number of elements labeled q and in each fiber. Suppose F p has a elements labeled with and b elements labeled with q. Additionally, suppose x of the elements that are labeled with have a cover in a separate fiber labeled with q and suppose y of the elements that are labeled with q cover an element in a separate fiber labeled with a . When performing jdt q , the x labels of in F p change to q and the y labels of q in F p change to . Observe that the application of jdt q may cause some labels of q and to change positions within F p . However, in Definition 3.1, jdt q prioritizes (1a) and (1c), so this might not occur. Because we know a label remains in its fiber after the first application of jdt q , the remaining applications of jdt q results in all labels above all labels of q in F p . Additionally, we can determine that there are a − x + y elements labeled and b + x − y labeled q in F p . After performing (jdt q ) for all fibers, we apply jdt →(q+1) to replace all labels of with q + 1, then reduce every label by 1. The result in F p is that we now have b + x − y elements labeled q − 1 and a − x + y elements labeled q.
To determine what happens when we apply ρ q−1 • JdtPro q−1 (f ), we begin by performing jdt →(q) (f ) and subtracting 1 from all unfrozen labels. F p will have a elements labeled with q − 1 and b elements labeled with q. Furthermore, we know that x of the elements that are labeled with q − 1 will have a cover in a separate fiber labeled with a q and that y of the elements that are labeled with q will cover an element in a separate fiber that is labeled with a q − 1. This means F p has a − x labels of q − 1 that are free and b − y labels of q that are free. Performing ρ q−1 switches these into a − x elements labeled with q and b − y elements labeled q − 1. Combining this with the x fixed labels of q − 1, we obtain b + x − y elements labeled q − 1. Similarly, with the y fixed labels of q, we obtain a − x + y elements labeled q. This matches the JdtPro(f ) case, allowing us to conclude that JdtPro(f ) = ρ q−1 • JdtPro q−1 (f ).  Before presenting the main result of this section, we first give an example demonstrating ρ q−1 • JdtPro q−1 and the result of Lemma 3.8. Example 3.9. Figure 8 shows an example of ρ q−1 • JdtPro q−1 being applied to the same P -strict labeling from Figure 7 and Example 3.2. To perform JdtPro 2 , we first freeze all labels that are greater than 2. In Figure 8, these frozen labels are colored blue. We then apply jdt →3 • (jdt 2 ) 5 • jdt 1→ (f ). Note that in Figure 8, we do not show applications of jdt 2 that do nothing. Following this, we subtract all unfrozen labels by 1. After this step, we have finished applying JdtPro 2 , so all labels are now considered unfrozen. We conclude by applying the Bender-Knuth involution ρ 2 . Observe that the resulting P -strict labeling in Figure 8 is identical to the P -strict labeling in Figure 7 obtained by applying JdtPro. Lemma 3.8 ensures that this will always be the case.
We proceed to the main theorem of this section, which states that P -strict promotion via jeu de taquin and P -strict promotion via Bender-Knuth toggles are equivalent. Our proof uses Lemma 3.8 and an inductive argument.
Proof. Let f ≤i with restriction R i denote the P -strict labeling f restricted to the subposet of elements with labels less than or equal to i. Observe that by Lemma 3.8, we have JdtPro . By induction, we know this holds for i = q − 1, yielding JdtPro(f ≤q ) = ρ q−1 • ρ q−2 • · · · • ρ 1 (f ≤q ), which is the desired result.
3.2. P -strict evacuation. Evacuation has been well studied on both standard tableaux and semistandard tableaux. In [3], Bloom, Pechenik, and Saracino provide explicit statements and proofs for several evacuation results on semistandard tableaux. We define evacuation on P -strict labelings and investigate which of those results can be generalized and which cannot.
Evacuation and dual evacuation have a special relation on rectangular semistandard Young tableaux. We generalize that relation here. Definition 3.12. Fix the following notation for the product of chains poset: be the antipode of (i 1 , i 2 , . . . , i k ).
, we obtain a new labeling by interchanging each label with the label of its antipode, then replacing each label i with q + 1 − i. Denote this new labeling as f + .
Proof. This follows from the definitions of evacuation and dual evacuation as a product of Bender-Knuth involutions.
Since P -strict labelings generalize both increasing labelings and semistandard Young tableaux, a natural aim would be to generalize results from these domains. Bloom, Pechenik, and Saracino found a homomesy result on semistandard Young tableaux under promotion [3, Theorem 1.1]. A natural generalization to investigate would be to P -strict labelings under promotion, where P is a product of two chains and = 2. We find that the result does not generalize due to several evacuation results failing to hold. We note below two statements on evacuation which do generalize and two examples showing statements that do not generalize. Proposition 3.16. Let P be a poset. For f ∈ L P ×[ ] (u, v, R q ), we have the following: Proof. Both parts rely only on the commutation relations of toggles (see Remark 1.24), and therefore follow using previous results on the toggle group. Figure 9 gives a counterexample. f Pro 7 (f ) Figure 9. By applying Pro 7 to the P -strict labeling f on the left, we obtain the P -strict labeling on the right. We see that these are not equal and so Pro q (f ) = f does not hold in general.

Applications of the main theorems to tableaux of many flavors
In this section, we apply Theorems 2.8 and 2.20 to the case in which P is a chain; in the subsections, we specialize to various types of tableaux. We translate results and conjectures from the domain of P -strict labelings to B-bounded Γ(P,R)-partitions and vice versa. 4.1. Semistandard tableaux. First, we specialize Theorem 2.8 to skew semistandard Young tableaux in Corollary 4.3. We relate this to Gelfand-Tsetlin patterns and show how a proposition of Kirillov and Berenstein, Corollary 4.6, follows from our bijection. Finally, we state some known cyclic sieving and homomesy results and use Corollary 4.3 to translate between the two domains.
We begin by defining skew semistandard Young tableaux.
Definition 4.1. Let λ = (λ 1 , λ 2 , . . . , λ n ) and µ = (µ 1 , µ 2 , . . . , µ m ) be partitions with non-zero parts such that µ ⊂ λ. Where applicable, define µ j := 0 for j > m. Let λ/µ denote the skew partition shape defined by removing the (upper-left justified, in English notation) shape µ from λ. A skew semistandard Young tableau of shape λ/µ is a filling of λ/µ with positive integers such that the rows increase from left to right and the columns strictly increase from top to bottom. Let SSYT(λ/µ, q) denote the set of semistandard Young tableaux of skew shape λ/µ with entries at most q. In the case µ = ∅, the adjective 'skew' is removed. In this and the next subsections, fix the chain [n] = p 1 p 2 · · · p n . We also use the notation n for the partition whose shape has n rows and columns.
Proposition 4.2. The set of semistandard Young tableaux SSYT(λ/µ, q) is equivalent to Proof. Each box (i, j) of a tableau in SSYT(λ/µ, q) corresponds exactly to the element (p i , j) in u . The weakly increasing condition on rows and strictly increasing condition on columns in SSYT(λ/µ, q) corresponds to the weak increase on fibers and strict increase on layers, respectively, in We now specify the B-bounded Γ(P,R)-partitions in bijection with SSYT(λ/µ, q). RecallB from Definition 2.7.
Proof. By Proposition 4.2, P -strict labelings L [n]×[λ 1 ] (u, v, R q ) with u and v as above are exactly semistandard Young tableaux of shape λ/µ with largest entry q, SSYT(λ/µ, q). Therefore, the first claim follows from Corollary 2.24, where a(p i ) = 1 and b(p i ) = q for all 1 ≤ i ≤ n. The second claim follows directly from Theorem 2.8. When P = [n], the lemma underlying our first main theorem is equivalent to a result of Kirillov and Berenstein regarding the correspondence between Bender-Knuth involutions on semistandard Young tableaux and elementary transformations on Gelfand-Tsetlin patterns. We define these objects below and then state their result, Corollary 4.6, in our notation. We then prove a more general result from which this follows, Theorem 4.8, as a corollary of our first main theorem.
where we consider a ij = ∞ if j < 1 and a ij = 0 if j > i + m.
We use the mechanism of our main theorem to prove Theorem 4.8, which yields the following result. We prove this corollary right before Remark 4.11.
To put this corollary in the language of our main theorem, we show that GT(λ,μ, q) is equivalent to A B (Γ([n], R)), where the restriction function R and the bounding function B are defined below. Definition 4.7. For any convex subposet P × [ ] v u and global bound q, let R be the (not necessarily consistent) restriction function on P given by R(p) = {0, 1, . . . , q + 1} for all p ∈ P , and let B be defined on Γ(P, R) as B(p, 0) = − u(p) and B(p, q) = v(p).
Thus the structure of Γ(P, R) consists of the chains (p, 0) (p, 1) · · · (p, q) and we have (p, k) (p , k + 1) whenever p P p and 0 ≤ k ≤ q − 1. As we will see in the proof, these covering relations provide the inequality conditions (2)  By generalizing semistandard tableaux to P -strict labelings, we are able to prove the equivariance result of Corollary 4.6 for any poset P . In this way, A B (Γ(P, R)) can be considered a generalization of Gelfand-Tsetlin patterns. and ρ k on L P ×[ ] (u, v, R q ) corresponds to τ k on A B (Γ(P, R)).
We first define the bijection map using the value j p k from Definition 2.13. Recall from Definition 2.12 that we consider the label f (p, i) to be in position i.  (Γ(P, R)) where Ψ(f )(p, k) = + 1 − j p k . We can treat Ψ(f )(p, k) as the number of positions j in the fiber F p such that f (p, j) is larger than k, where we consider f (p, i) > k in the positions + 1 − v(p) ≤ i ≤ for which f is not defined. Figure 11 for an example of the map Ψ. Proof. We begin by verifying that Ψ(f ) ∈ A B (Γ(P, R)). For 1 ≤ k ≤ q, (p, k) (p, k − 1). Since f is weakly increasing on fibers, we have Ψ(f )(p, k) ≤ Ψ(f )(p, k − 1), as there must be at least as many positions greater than k − 1 as are greater than k. If p P p and 0 ≤ k ≤ q − 1, then (p, k) Γ(P,R q ) (p , k + 1). Since there are Ψ(f )(p, k) positions greater than k in f (F p ), there must be at least as many positions greater than k + 1 in f (F p ) in order to accommodate those values in f (F p ), as f is strictly increasing on layers. Thus Ψ(f )(p, k) ≤ Ψ(f )(p , k + 1), so Ψ(f )(p, k) respects all covering relations in Γ(P, R). Moreover, Ψ(f )(p, 0) = − u(p) since the first position greater than zero is at f (p, u(p) + 1) for all p, and Ψ(f )(p, q) = v(p) since the only positions considered greater than q are those after the end of the fiber. Thus Ψ(f ) ∈ A B (Γ(P, R)).
Let k ∈ R q (p) * . If (p, k) covers and is covered by the same elements in Γ(P, R q ) as in Γ(P, R), then we are done, so we will consider the cases in which these covers differ. Suppose k 1 > k + 1 and either (p, k) (p, k 1 ) in Γ(P, R q ) or there exists p P p such that (p, k) (p , k 1 ). In each case, by definition of Γ, k + 1 / ∈ R q (p) * so, by Lemma 2.23, (p, k) is fixed in A B (Γ(P, R q )) and therefore in A B (Γ(P, R)). Now suppose k 1 < k − 1 and either (p, k) (p, k 1 ) or there exists p P p such that (p, k) (p , k 1 ). In the first case, σ(p, k 1 ) = σ(p, k 1 ) = σ(p, k − 1). In the second case, k − 1 / ∈ R q (p ), otherwise we would have (p, k) (p , k − 1), so σ(p , k 1 ) = σ(p , k 1 ) = σ(p , k − 1). In both cases where the covers in Γ(P, R q ) differ from Γ(P, R), the minimum value of the upper covers and the maximum value of the lower covers of (p, k) is unchanged between A B (Γ(P, R q )) and A B (Γ(P, R)). Thus, τ (p,k) (σ)(p, k) = τ (p,k) (σ)(p, k).
By the above, τ k on A B (Γ(P, R)) is equivalent to τ k on A B (Γ(P, R q )). Thus, by Lemma 2.11, τ k on A B (Γ(P, R)) corresponds to ρ k on L P ×[ ] (u, v, R q ).
In the following proof of the Kirillov and Berenstein result, we consider a Gelfand-Tsetlin pattern as a parallelogram-shaped array {a ij } 0≤i≤q,1≤j≤n with the same properties as Definition 4.4.
In the case where µ = ∅ and λ is a rectangle, Corollary 4.3 specializes nicely. We now discuss a cyclic sieving result of B. Rhoades on rectangular semistandard Young tableaux and its translation via Corollary 4.12.
Definition 4.13 ([27]). Let C be a finite cyclic group acting on a finite set X and let c be a generator of C. Let ζ ∈ C be a root of unity having the same multiplicative order as c and let g ∈ Q[x] be a polynomial. The triple (X, C, g) exhibits the cyclic sieving phenomenon if for any integer d ≥ 0, the fixed point set cardinality |X c d | is equal to the polynomial evaluation g(ζ d ).
Theorem 4.14 ([28, Theorem 1.4]). The triple (SSYT( n , q), Pro , X(x)) exhibits the cyclic sieving phenomenon, where Proof. This follows from Theorem 4.14 and Corollary 4.12. Note that X(x) is MacMahon's generating function for plane partitions which fit inside a box having dimensions by n by q − n. These are in simple bijection with A ([n] × [q − n]).
We now turn our attention toward several homomesy results. Rather than present the most general definition, this definition is given for actions with finite orbits, as this is the only case we consider. where |O| denotes the number of elements in O. If such a c exists, we will say the triple is c-mesic.
We state two known theorems below and prove their equivalence as a corollary of Theorem 2.8.

Flagged tableaux.
In this section, we first specialize Theorem 2.8 to flagged tableaux and use this correspondence to enumerate the corresponding set ofB-bounded Γ(P,R)-partitions. Then, we state some recent cyclic sieving and new homomesy conjectures and use Theorem 2.8 to translate these conjectures between the two domains.
correspond to semistandard Young tableaux whose entries in row i are restricted above by b i , which is exactly FT(λ/µ, b).
Remark 4.25. Flagged tableaux are enumerated by an analogue of the Jacobi-Trudi formula due to I. Gessel and X. Viennot [18] with an alternative proof by M. Wachs [38]. Thus the bijection of Corollary 4.24 allows one to translate this to enumerate A B (Γ([n], R b )).
In the rest of this subsection, we apply Corollary 4.24 to some specific sets of flagged tableaux, obtaining Corollaries 4.28 and 4.41 along with further corollaries and conjectures. Our first corollary involves the triangular poset from the following definition. This poset is isomorphic to the Type A n positive root poset from Coxeter theory. Though this algebraic interpretation is what has generated interest surrounding this poset, we will not need it here.  n by (p i , k) → (i, n − k + i). Since i ≤ k ≤ 2i − 1 we have n − i + 1 ≤ n − k + i ≤ n + 1, so the above map is a bijection to {(i, j) | i + j > n}. Because (i, j) (i , j ) ∈ [n] × [n] if and only if i = i and j + 1 = j or i + 1 = i and j = j , the covers of (p i , k) in Γ([n], R b ) correspond exactly to the covers of (i, n − k + i) in n . Thus Γ([n], R b ) and n are isomorphic as posets.  We then obtain the following as a corollary of this theorem and Corollary 4.28.
Thus, Corollary 4.28 implies the equivalence of this conjecture and the following. We conjecture the following homomesy statement (Conjecture 4.35), which was proved in the case = 1 by S. Haddadan [20,21]. Definition 4.33. We say a poset P is ranked if there exists a rank function rk : P → Z such that p 1 P p 2 implies rk(p 2 ) = rk(p 1 ) + 1.
For the following conjecture, we use the rank function of n defined by rk(p) = 0 if p is a minimal element.
Conjecture 4.35. The triple A ( n ), TogPro, R is 0-mesic when n is even and 2 -mesic when n is odd.
Using Sage [35], we have checked this conjecture for n ≤ 6 and ≤ 3. We have also verified that a similar statement fails to hold for the Type B/C case when n = 2 and = 1, and the Type D case when n = 4 and = 1.
Note this is a set of flagged tableaux with different shape and flag but the same cardinality as the flagged tableaux in Corollary 4.28, the same conjectured cyclic sieving polynomial, and a different order of promotion. The case = 1 follows from a result of S.P. Eu and T.S. Fu [14] on cyclic sieving of faces of generalized cluster complexes, but for > 1 this conjecture is still open.
We can translate this conjecture to rowmotion on P -partitions with the following corollary of Theorem 2.8. Recall Definition 3.12, which specifies notation for [a] × [b].

Symplectic tableaux.
We begin by defining semistandard symplectic Young tableaux, following the conventions of [6].
We now specify theB-bounded Γ(P,R)-partitions in bijection with Sp(λ/µ, 2q). RecallB from Definition 2.7.  There is also a hook-content formula for symplectic tableaux, due to P. Campbell and A. Stokke [6]. They proved a symplectic Schur function version of this formula, but we will not need that here. 2q + r λ (i, j) h λ (i, j) where h λ (i, j) is the hook length h λ (i, j) = λ i + λ t j − i − j + 1 and r λ (i, j) is defined to be We use this formula to enumerate symplectic tableaux of staircase shape, finding a particularly simple formula. Proof. This follows from Theorem 4.47 above. For λ = sc n = (n, n − 1, . . . 1), we have λ i = λ t i = n − i + 1. First, we calculate the product of the numerator, where we always take (i, j) ∈ [λ], i.e. 1 ≤ i ≤ n and 1 ≤ j ≤ n − i + 1. In the rest of this subsection, we apply Corollary 4.45 to staircase-shaped symplectic tableaux, obtaining Corollaries 4.50 and 4.51. This involves the poset in the following definition. This poset is isomorphic to the dual of the Type B n positive root poset. As before, we will not need this algebraic motivation here. See Figure 14. We obtain the following correspondence, as a corollary of our main results.
Finally, we determine these upper and lower bounds on the label of any element (i, j) ∈ n by determining the corresponding bounds on the label σ(p i , k) where σ ∈ A B (Γ([n], R 2n a )) and (p i , k) ∈ Γ([n], R 2n a ) \ dom(B). For the fixed elements (p i , min R 2n a (p i ) * ) we haveB(p i , min R 2n a (p i ) * ) = n − u(p i ) = n, so these elements induce an upper bound of n on all σ(p i , k). Next, the fixed elements (p i , max R 2n a (p i ) * ) = (p i , 2n) induce a lower bound v(p i ) = i − 1 on all σ(p i , k) and an equivalent upper bound on σ(p i , k ), where (p i , k ) < (p i , 2n), which is the case whenever i < i and k ≥ 2n − (i − i ). Therefore, a generic σ(p i , k) is bounded below by i − 1 and above by at most n and, if k = 2n − (i − i) for any i < i ≤ n, then σ(p i , k) is bounded above by i − 1. Translating to A n ( n ), σ(i, j) = σ(p i , 2n − 1 + i − j) (we keep the notation σ due to the equivalence shown above) so σ(i, j) is bounded below by i−1 and above by at most n. We have 2n−1+i−j = 2n−(j +1−i), so σ(i, j) is bounded above by j for 1 ≤ j ≤ n − 1. Thus, if δ(i, j) = min(j, n) and (i, j) = i − 1, then A B (Γ([n], R 2n a )) is equivalent to A δ ( n ). The corresponding (δ, )-bounded n -partition is given on the right, shown as the equivalent element of A B (Γ( [3], R 6 a )). Here, the poset element (p 1 , 5) ∈ Γ([3], R 6 a ) corresponds to (1, 1) ∈ n , (p 1 , 4) corresponds to (1,2), and so on.
The corollary below follows directly from Corollaries 4.48 and 4.50.
It would be interesting to see whether one can find a set of symplectic tableaux that exhibit the cyclic sieving phenomenon with respect to promotion. A nice counting formula is generally a necessary first step.