Some Instances of Homomesy Among Ideals of Posets

Given a permutation $\tau$ defined on a set of combinatorial objects $S$, together with some statistic $f:S\rightarrow \mathbb{R}$, we say that the triple $\langle S, \tau,f \rangle$ exhibits homomesy if $f$ has the same average along all orbits of $\tau$ in $S$. This phenomenon was noticed by Panyushev (2007) and later studied, named and extended by Propp and Roby (2013). After Propp and Roby's paper, homomesy has received a lot of attention and a number of mathematicians are intrigued by it. While seeming ubiquitous, homomesy is often surprisingly non-trivial to prove. Propp and Roby studied homomesy in the set of ideals in the product of two chains, with two well known permutations, rowmotion and promotion, the statistic being the size of the ideal. In this paper we extend their results to generalized rowmotion and promotion together with a wider class of statistics in product of two chains . Moreover, we derive some homomesy results in posets of type A and B. We believe that the framework that we set up can be used to prove similar results in wider classes of posets.


Introduction
Consider a poset P, and let J(P) be the set containing all of the order ideals in P. Various bijections can be defined mapping J(P) to itself. Among these mappings, the rowmotion (Definition 1) operation has been studied widely by combinatorialists [6,17,5], and under various names (Brouwer-Schrijver map [3], the Fon-der-Flaass map [15], the reverse map [10], and Panyushev complementation [1]). Another mapping called promotion (Definition 2.6) has been defined in analogy to rowmotion. Propp and Roby [14] were interested in studying the orbits these bijections introduce on J(P) and the statistics that are preserved along these orbits. They introduced a concept they called homomesy (Definition 2.8) which is defined on a triple consisting of a set S, a bijection on S, and a statistic on S, that is, a function from S to R. They studied homomesy of promotion and rowmotion on J(P), where P is a product of two chains.
Striker and Williams [17] generalized the definition of rowmotion and promotion (see Definitions 2.5 and 2.7) and proved interesting qualities of their orbit structures on the so called rc-posets.
In Section 2, we will provide the formal definition of all the above concepts, along with the statements of our main results. Our proofs are based on an orbit preserving bijection between the order ideals of a poset and sequences of natural numbers. In Section 3, we will present this bijection, and finally the proofs will appear in Section 4.

Our Contribution
Following the work of Propp and Roby [14] and Striker and Williams [17], we study rowmotion, promotion and their generalizations on J(P) where P is one of the following three posets: the product of two chains, the upper triangle of this product, or the right side triangle. We introduce a new bijection on the above sets called comotion (Definition 2.9). The main ingredient of the function comotion is a map on order ideals known as the toggle map (Definition 2.1). In Theorems 2.3, 2.4, and 2.5 we characterize a wide class of functions from J(P) to the real numbers which exhibit homomesy along the orbits of comotion.
Our results generalize Propp and Roby's results [14] in three ways: (1) Comotion captures rowmotion and promotion and our results reproduce results of Propp and Roby [14], and generalize them. (2) Our results are not restricted to the order ideals in the product of two chains but also the upper triangle of the product, and the right side triangle. (3) We introduce a set of functions such that any linear combination of them will constitute a homomesic static in our setting, and we observe that the functions studied by Propp and Roby can be constituted using linear combinations of these functions.
At the end of this introduction, we would like to mention that the operation "winching" (Definition 3.1 ) and the Theorems 3.1, 3.2, 3.3 are presented here to aid in understanding comotion on order ideals. In addition, since they are neat examples of homomesy, they may be of independent interest.

Previous Work, Definitions and Statements of Results
Definition 1. Given a poset P on the elements of set S, and an order ideal I ∈ J(P), the rowmotion of I, denoted Φ(I), is defined to be the order ideal generated by the minimal elements in S−I. and an element x ∈ S, the toggle map σ x : J(P) → J(P) is defined by: (1) Cameron and Fon-der-Flaass [4] also noted that each toggle is an involution and gave a criterion for when toggles commute.
For all x ∈ S and I ∈ J(P), σ 2 x (I) = I. If x, y ∈ S, and x does not cover y, and y does not cover x, we have σ x • σ y (I) = σ y • σ x (I).
A linear extension (x 1 , . . . , x n ) of P is an indexing of the elements of P satisfying x i < P x j implies i < j. Cameron and Fon-der-Flaass [4] observed that the rowmotion operation coincides with a series of compositions of the toggle map as follows: 4]). Given an arbitrary I ∈ J(P) and linear extension ( and we say (i 1 , j 1 ) (i 2 , j 2 ) if and only if i 1 i 2 and j 1 j 2 . In this paper, we are interested in bijections on J(Q a,b ), as well as J(U a ) and J(L a ) where U a and L a are subsets of Q a,a and defined as: The upper triangle lattice U a and the right side triangle lattice L a are two sublattices of Q a,a defined respectively as, See Figure 1 for examples of upper triangle and right side triangle lattices. We use the following notation throughout: Let P be one of Q a,b , U a or L a . By saying (i, j) ∈ P we are referring to the element in [a] × [b] with coordinates i and j. By saying x = (i 1 , j 1 ) P y = (i 2 , j 2 ) we mean x is less than or equal to y in P, and we may drop the subscript if no confusion would be introduced.
We call Q a,b the square lattice or the product of two chains, U a the upper lattice and L a the left lattice. Among combinatorists U a is also known as the root poset of type A a , and L a as the minuscule poset of type B a or D a+1 (See [15]).
We employ the following terminology: For any arbitrary I ∈ J(P), • We call the set of all points (i, j) ∈ P with constant i+j a rank; R c (I) = {(i, j) ∈ I | i+j = c}.
• We call the set of all points (i, j) ∈ P with constant i−j a file; • We call the sets of all points (i, j) ∈ P with constant i a column; When P is clear from the context, we write R c = R c (P), F c = F c (P) and C c = C c (P).
See Figure 1 for examples of rank, file and column in different lattices.
We can now define toggling for the above sets.
Definition 2.4. Consider I ∈ J(P). Take an arbitrary c, and let S be one of R c or F c . Picking an arbitrary indexing of the elements of S, Note that in the above definition, no two elements x i , x j of S constitute a covering pair, thus by Proposition 2.1 σ S is well defined.
Striker and Williams studied the class of so-called rc-posets, whose elements are partitioned into ranks and files 1 . Here, we are interested in Q a,b , U a or L a which are all rc-posets. The following definitions are from [17], restricted to the product posets of interest to us. Again, by saying P we mean one of Q a,b , U a or L a : Definition 2.5 ( [17]). Consider a poset P. Let ν be a permutation of {2, . . . , a+b}. We define Φ ν to be σ R ν(a+b−1) • σ R ν(a+b−2) • · · · • σ R ν (1) .
We now present another bijection on J(P ) which has been studied in [17,14]: Definition 2.6. The mapping promotion is a permutation ∂ : J(P) → J(P), defined on I ∈ J(P) as: As with rowmotion, Striker and Williams [17] define a generalized version of promotion.
Like with rowmotion, for any permutation ν on files of any rc-poset P, ∂ ν will partition J(P) into orbits. Again, Striker and Williams [17] showed that regardless of which ν we choose, J(P) will be partitioned into the same orbit structure by ∂ ν . Moreover, the orbit structures of ∂ ν and Φ ω are the same for any two permutations ν and ω: Note that the above theorem holds in particular for Q a,b which is a rc-poset.
In 2013, Propp and Roby introduced a phenomenon called homomesy [14]. Propp and Roby discussed some instances of homomesy by studying the actions of promotion and rowmotion on the set J(Q a,b ). After Propp and Roby's paper, homomesy has attracted the attention of many combinatorialists [5,8,9,2,7], and it is defined as follows: 14]). Consider a set S of combinatorial objects. Let τ : S → S be a permutation that partitions S into orbits, and f : S → R a statistic of the elements of S. We call the triple S, τ, f homomesic (or we say it exhibits homomesy) if and only if there is a constant c such that for any τ -orbit O ⊂ S we have Equivalently, we can say f is homomesic, c-mesic or it exhibits homomesy in τ -orbits of S.
The following is easily shown from the definition of homomesy. The following theorem is a result of Propp and Roby [14]: 14]). Consider f : J(Q a,b ) → R defined as follows: for all I ∈ Q a,b , f (I) = |I|. Let ∂, Φ : J(Q a,b ) → J(Q a,b ) be the rowmotion and promotion operation. The triples J(Q a,b ), ∂, f and J(Q a,b ), Φ, f exhibit homomesy.
We now present definition of comotion, which generalizes rowmotion and promotion. Later in this paper, we generalize Theorems 2.2 and 2.1 for comotion. Definition 2.9. For any permutation ν of [a], we define the action comotion, T ν : J(P) → J(P), for any I ∈ J(P) as follows: Remark 2. Note that by applying Proposition 2.1 inductively on the posets Q a,b , U a and L a , the action of promotion coincides with T (a,a−1,...1) and rowmotion coincides with T (1,2,...a) .
We are now ready to state the main theorems of this paper, which are Theorems 2.3, 2.4, 2.5. We leave the proof of these theorems to Sections 3 and 4. Before stating the main results we need to define the following functions which constitute building blocks of the functions for which we observe and prove homomesy: Definition 2.10. Let C i be a column in Q a,b , L a or U a . For 1 i a and 1 j b, we define g i,j , s i,j , d i,j , and s j and κ j for I ∈ J(P) as follows: • For any 1 i a and 1 j b, • For any 1 i a and 1 j b, • For any 1 i a and 1 j b, • For any 1 j b, • For any 1 j b, In Theorems 2.3, 2.4, and 2.5 and Corollaries 2.1, 2.2, and 2.3, let a be an arbitrary natural number and ν a permutation on [a], and T ν as in Definition 2.9: 2. For any 1 i a and 1 j b, the following triples are homomesic.
Theorem 2.3 introduces a new family of permutations having the same orbit structure as Φ and ∂; hence, it generalizes Theorem 2.1. By Remark 2 any linear combination of the above functions is also homomesic, thus, Theorem 2.3 also generalizes Theorem 2.2 by introducing a wide class of permutations and statistics whose triples with J(Q a,b ) exhibit homomesy.
The proof of Theorem 2.3 is involved; we skip it in this section and will present it later in the paper. We will provide a roadmap to the proof after stating the following two theorems about homomesy in J(U a ) and J(L a ).
1. The orbit structure of T ν on J(L a ) is independent of choice of ν.
The proofs of Theorem 2.3, 2.4, and 2.5 are postponed to the next sections. The heart of these proofs is the correspondence between comotion and a function which we define in Section 3 called winching (See Definition 3.1). We define three variations of winching on sequences of increasing numbers which correspond to comotion in J(Q), J(U), and J(L).
A detailed presentation of these correspondences, together with formal definitions is given in Section 3. After introducing these definitions, we show that there is a natural equivariant bijection between the set of order ideals under comotion and the set of increasing sequences under winching. These correspondences simplify our theorems, and in fact in Section 3 we have Theorems 3.1, 3.2, and 3.3 which are in the context of winching and equivalent to Theorems 2.3, 2.4, and 2.5 of this section. Finally, in Section 4, by proving the theorems of Section 3 we complete the proofs of all the theorems in this paper.
We continue this section by employing the above theorems to find homomesy of some well known functions in the orbit structure produced by comotion in J(Q a,b ), J(U a ), and J(L a ). For example one important function also studied by [14] is size of an order ideal, i.e. sz(I) = |I|. Here is a list of functions we consider: For any arbitrary . We call CA x the central antisymmetry function with respect to x. The central antisymmetry function characterises the presence of one and only one of x and its antipode in an order ideal.
3. Rank-alternating cardinality. Consider an arbitrary I ∈ J(P). We denote the rank-alternating cardinality of I by RAC(I) and we define it as RAC(I) = For any choice of ν, the following homomesy results are concluded from Theorems 2.3, 2.4, and 2.5, and the fact that any linear combination of homomesic functions is homomesic (Remark 2): In both cases sz is a linear combination of s i . So, using Theorems 2.3 and 2.5 we will have the result.
By Equations 8 and 9, we have CA Proof. We will first consider the case when I ∈ J(Q a,b ). In this case we have:

Comotion, winching and their correspondence
In the previous section, we defined the action of comotion on the set of order ideals of a poset. In this section, we define winching and show a correspondence between winching on increasing sequences and comotion on J(Q a,b ). Then, we define winching with lower bounds and winching with zeros. The former corresponds to comotion on J(U a ) and the later corresponds to comotion on J(L a ).
Definition 3.1. Let S k,m be the set of all k-tuples x = (x 1 , . . . , x k ) satisfying 0 < x 1 < x 2 < · · · < x k < m+1. We define the map W i : S k,m → S k,m , called winching on index i, by W i (x) = y = (y 1 , y 2 , . . . , y k ), where y j = x j for i = j, and We assume that always x 0 = 0 and x k+1 = m+1.
For arbitrary ν = (ν 1 , ν 2 , . . . , ν k ) a permutation of [k] we define W ν to be the function We now show that winching is equivalent to toggling by columns on J(Q a,b ).
Example 3.2. Figure 2 illustrates an application of Lemma 3.1 to an order ideal in J(Q 4,5 ). Note that as Lemma 3.1 suggests we have: I 2 = σ C 3 (I 1 ) and I 3 = σ C 4 (I 2 ). Equivalently we have: W 3 (α(I 1 )) = α(I 2 ) and W 4 (α(I 2 )) = α(I 3 ). In what follows we define two variations of winching: winching with lower bounds and winching with zeors. Then, we define two equivariant bijections from J(U a ) and J(L a ) to sequences of numbers. Finally, we show in Lemmas 3.2 and 3.3 that these bijections map toggling to winching with lower bounds in J(U a ) and to winching with zeros in J(L a ) (see Figure 3). The following variation of winching is called winching with lower bounds, and we show in Lemma 3.2 that it corresponds to comotion on J(U a ).
Since comotion is a sequential composition of toggles applied to different columns, for any choice of ν we have the following corollaries: Corollary 3.1. The bijection α introduced in Lemma 3.1 satisfies the following property: α(T ν (I)) = W ν (α(I)).
We now proceed to the theorems of this section, which discuss the occurrence of homomesy in the winching setting and are in correspondence with Theorems 2.3, 2.4 and 2.5 stated in the previous section.
We consider the following functions: • Let g i,j : S k,m → R, 1 i k and 1 j m be defined as follows: • For an arbitrary 1 j m, let f j : S k,m → R be defined by: In Theorems 3.1, 3.2 and 3.3 ν is an arbitrary permutation on [k] or [n]. The proofs of these theorems will be presented in Section 4: 1. W m ν (x) = x for all x ∈ S k,m .
2. We observe homomesy in the following triples: (a) For each 1 i, j k, the function d i,j = g i,j − g k+1−i,m+1−j is zero-mesic.
(b) For any 1 j m, the triple S k,m , W ν , f j is homomesic and the average of f j along W ν orbits is k/m.
We will prove the above Theorem in Section 4.1.
Remark 3. The orbit structure that winching produces on the set S k,m is the same as the orbit structure for rotation acting on the set of 2-colored necklaces with k white beads and m − k black beads, and hence independent of choice of ν. (The orbit structure of necklaces is a classical problem in combinatorics, and it is known as Pólya's Theorem [11].) Theorem 3.2 (Homomesy for winching with lower bounds). Consider W ν : S k,n → S k,n with lower bounds (l 1 , l 2 , . . . , l k ). Let [i, k+n] = i, i + 1 . . . k+n for an arbitrary i k + n, and assume f : [k+n] → R is a function that has the same average in all [l i , k + n]. Let g : S k,n → R be defined as, Then, the triple S k,n , W ν , g exhibits homomesy.
We will prove the above Theorem in Section 4.2.
Proof of Theorem 2.4. Given the bijection in Corollary 2.2, Theorem 2.4 is concluded from Theorem 3.2.
2. The triple S n , WZ ν , f j (defined in Definition 3.4) is homomesic. Furthermore, the average of f j along WZ ν -orbits is 1 2 . We will prove the above theorem in Section 4.3.
Proof of Theorem 2.5. Given the bijection in Corollary 2.3, Theorem 2.5 is concluded from Theorem 3.3.

Proofs
In this section we will prove Theorems 3.1, 3.2, and 3.3. The concepts of a tuple board and a snake (Definitions 4.1 and 4.2) play a key role in understanding the orbit structure and homomesy in winching.
For fixed k and an arbitrary permutation on [k], namely ν = (ν 1 , ν 2 , . . . , ν k ), let F ν be one of W ν , W ν or WZ ν . Let S = S k if F ν = WZ ν and S = S k,m otherwise. We define a tuple board as follows: Figure 4).
Remark 4. Note that since F is a permutation, there is always an n satisfying F ν n+1 (x) = x. In fact, a tuple board contains an orbit of winching, thus it is convenient to think of it as a cylinder. When referring to row numbers in a tuple board they are understood modulo n.
Notice that any cell in a tuple board contains a number from the set {0, 1, 2, . . . , m}.
In any tuple board we partition the cells having non-zero elements to maximal sequences of adjacent cells of consecutive numbers. We call any of such partitions a snake. More precisely: and ν(j) > ν(j + 1).
(17) Depending on the choice of F v , tuple boards will be filled with snakes differently. The proofs of Theorems 3.1, 3.2 and 3.3 are based on understanding the configuration of snakes in tuple boards. Figure 5 illustrates this by presenting three examples of snake configurations corresponding to winching, winching with lower bounds and winching with zeros. We formalize these observations in proofs of Theorems 3.1, 3.2 and 3.3 presented in the following subsections.
The snake map is (0, 1, 2, 1, 1). (0, 1, 4, 1, 1)  To see that two different snakes are either the same or have no intersection, assume we have s = (s 1 , s 2 , . . . s k ) and s = (s 1 , s 2 , . . . , s k ). First assume s H(s) = s H(s ) let l > 1 be the minimum index such that s l = s l . Consider the pair i, j such that T (i, j) = l, T (i + 1, j) = l + 1 and also either T (i, j + 1) = l + 1 or T (i + 1, j + 1) = l + 1. However, in this case by Definition 4.2 we have s l = s l = T (i + 1, j) which is a contradiction. Now assume s H(s) = T (1, r), s H(s ) = T (1, r ) and without loss of generality assume that r > r and ν = (1, 2, . . . k). Let l > 1 be the minimum index such that s l = s l = T (i, j). But this is contradiction because l = j + 1 + (i − r) and l = j + 1 + (i − r ) which is only possible if r = r . In the next lemma we show that two consecutive snakes maps are related by the left shift operator. This is essential to the proof of Theorem 3.1.

Lemma 4.2.
Consider an arbitrary tuple board T , and let s < s be two consecutive snakes in it. Assume c = S(s) and c = S(s ) are the two corresponding snake maps. We have c = H(c ). c = (c 1 , c 2 , . . . c k ) and c = (c 1 , c 2 , . . . c k ). We first show that for any 2 j k, c j−1 = c j . Then having that both s and s have length k we also conclude c k = c 1 . For any arbitrary 2 j k, using induction we show that if f = min{i | T (i, j − 1) ∈ s}, l = max{i | T (i, j − 1) ∈ s}, and f = min{i | T (i, j) ∈ s }, l = max{i | T (i, j) ∈ s }, then f = f + 1 and l = l + 1. For j = 2 this is clear by the fact s and s have no intersection.

Proof. Take
By the induction hypothesis assume that we have c 1 = c 2 , c 2 = c 3 , . . . , c j−1 = c j . We want to prove that c j = c j+1 . Let q and q be, respectively, the length of s truncated within columns 1 through j and the length of s truncated within columns 1 through j −1. From the inductive hypothesis, we conclude q = q + c 1 > q . Thus if c j > c j+1 then either length of s is less than length of s which is a contradiction or s and s intersect which is also a contradiction. If c j < c j+1 then there is a gap between s and s but this is also a contradiction. Thus, we have c j = c j+1 (see Figure 6). Proof. Since the left shift operator satisfies H k (c) = c, the placements of the ith snake and the (i + k)th snake will be the same. Thus, the rows in the tuple board will be repeated after the kth snake appears in the tuple board.  Proof. For any r, we construct a mapping from {∪ r t=1 T t } to {∪ k t=k−r+1 T t }. Consider a number x in T r . Let it be the lth element in T r , covered by a snake having snake map p = (c 1 , c 2 , . . . , c r , . . . , c k ). Consider the snake with snake map p = (c r+1 , . . . , c 1 , c 2 , . . . , c r ). Let y be the  Having the above mapping, we know there is also a one-to-one mapping in {∪ r t=1 T t } → {∪ k t=k−r+1 T t } and also in {∪ r−1 t=1 T t } → {∪ k t=k−r T t }. Hence, there exists F : T r → T k+1−r satisfying F(x) = m+1−x.
Proof of Theorem 3.1, Part 2. Considering any m × k tuple board T , Corollary 4.1 shows that T is totally covered by k snakes. Therefore, each element 1 i m appears k times in the tuple board and therefore the average of f i as defined in Definition 3.4 is independent of the orbit and equal to k/m.
Fix an arbitrary number j and column i. Lemma 4.3 shows that the number of occurrences of number j in column i is equal to the number of occurrences of m−j+1 in column

Proof of Theorem 3.2
In this subsection we prove Theorem 3.2. Remember the definitions of tuple board, snake, snake map and the correspondence to the action of winching with lower bounds.
Recall Lemma 4.1, and that if s is a snake in a tuple board of winching then H(s) = 1 and T (s) = m. In the following lemma we will show that if s is a W l ν -snake with lower bounds l = (l 1 , l 2 , . . . , l k ), then H(s) is equal to some l i and T (s) = m (see Figure 5).

Lemma 4.4.
For an arbitrary x ∈ S k,m , consider a tuple cylinder T (x) corresponding to application of the action of W l ν with lower bounds l = (l 1 , l 2 , . . . , l k ). For any snake s in this tuple board we have H(s) ∈ l, and T (s) = m.
Proof. Consider a snake s in T . If H(s) is in column i and H(s) l i then s is not maximal. If T (s) m, then s is not maximal either. Thus we have H(s) ∈ l, and T (s) = m.
Proof of Theorem 3.2. Any tuple cylinder corresponding to W l ν can be partitioned into snakes. Any snake starts with some l i ∈ l and ends in m. Thus, if f is a function having the same average on all the numbers contained in any snake, it has the same average over all the elements in the tuple cylinders. Therefore, we will have the result.

Proof of Theorem 3.3
In this section, we will prove Theorem 3.3. Consider x ∈ S n and the action of WZ ν for some arbitrary permutation ν of [n], and consider T = T B(x). In what follows, we will prove Lemmas 4.5, 4.6, 4.7 which correspondingly show: In any tuple board of winching with zeros (1) any snake has length n, (2) the snake maps evolve through a bijection called "crawling" (Definition 4.6), and in tuple board of winching with zeros there is one and only one row between heads of any two consecutive snakes. Finally, (3) crawling has orbit size n.
The proof of the following lemma is similar to the proof of Lemma 4.1, thus, we omit it. We now need to characterize the snake maps in winching with zeros. We first present the definition of crawl (Definition 4.6) and then we present Lemma 4.6.  Proof. Consider a snake s whose head is at row i in T and assume that in its snake map, the initial segment c 1 , c 2 , . . . , c k 0 is all zeros, and this segment is followed by c k 0 +1 , . . . , c k 1 all ones. These two segments are then followed by c k 1 +1 , . . . all greater than 1. This means that x i = (x 1 , x 2 , . . . , x n ) and x 1 , x 2 , . . . , x k 0 = 0, for i k 1 − k 0 x i+k 0 = i, and x k 1 +1 k 1 − k 0 + 1. Thus applying winching with zeros to x i , we take x = x i+1 . Thus we have x i+1 = (x 1 , x 2 , . . . , x n ) satisfying x i = 0 for any i < k 1 and x k 1 = x k 1 + 1. Applying winching once again, and letting x i+2 = x , we have x i+2 = (x 1 , x 2 , . . . , x n ) satisfying x i = 0 for any i < k 1 − 1 and x k 1 −1 = 1. Thus, H(s ) will be in row i + 2 and in c we will have: c 0 , c 1 , . . . , c k 1 −1 = 0.
Note that in the action of crawling the initial segment which is all zeros and ones is mapped to a sequence of all zeros and the rest of the elements in the sequence evolve through the left shift operator. We have so far proved that the initial segment is mapped to all zeros. In remains to prove that c k 1 = c k 1 +1 − 1 and for any i > k 1 , c i = c i+1 .
Similar to proof of Lemma 4.2 we can conclude c k 1 = c k 1 +1 − 1 because the tuple board is covered by the snakes of length n, and that there is one row between H(s) and H(s ). Since after this point there is no gap between the snakes we have: for any i > k 1 , c i = c i+1 . To complete the proof, note that c n and c n should always be such that the entire snake map sums to n, thus we have: c n = n − n−1 i=1 c i (see Figure 7). The next lemma which is Lemma 4.7 completes the characterization of W Z−snakes.
In particular it shows that the orbit size of crawl is n. The proof of this lemma is involved, and we prove it through a series of definitions and lemmas.  Proof of Theorem 3.3 Part 1. From Lemma 4.6 we know that head of snakes are located in alternating rows. By Lemma 4.7 we know that each snake gets back to itself after n crawls. Thus, in a tuple board of winching with zeros the 1st and 2nth rows are identical, i.e. W Z 2n (x) = x.
Proof of Theorem 3.3 Part 2. Consider a tuple board of winching with zeros. In the previous part of this theorem we proved that this board is a 2n × n board. In Lemma 4.6 we showed that head of snakes appear alternatively in rows. Thus, the number of snakes is n, and we conclude that half of the tuple board is filled with zeros. In addition, since there are n snakes in any tuple board and in any snake j appears once and only once, each nonzero number will appear n times in the tuple board, that is, the average of f j = 1 2 for each j.
We now proceed to prove Lemma 4.7. To this end, we introduce two bijections in Definitions 4.7 and 4.8. In Lemma 4.3 we show that the bijection F (Definition 4.7) preserves the orbit structure of winching with zeros.  Proof. Assume F(x) = F(y) = w, and let j be the smallest index where w j = 1. We have x 1 = y 1 = j. The next nonzero index will determine that x 2 = y 2 and likewise, we can verify that all entries of x and y are equal.   Proof. Consider an arbitrary c = (c 1 , . . . , c n ) ∈ M n . Let's say we have c 1 = · · · = c k−1 = 0, and c k is the leftmost nonzero element in c. Consider the set A = {a 1 = c k , a 2 = c k +c k+1 , . . . , a n−k = n i=k c i }. Let C (c) = c and b = (b 1 , . . . , b n ), the binary word representing A. In other words for all a, a ∈ A if and only if b a = 1.