Homomesy in products of two chains

Many invertible actions $\tau$ on a set ${\mathcal{S}}$ of combinatorial objects, along with a natural statistic $f$ on ${\mathcal{S}}$, exhibit the following property which we dub \textbf{homomesy}: the average of $f$ over each $\tau$-orbit in ${\mathcal{S}}$ is the same as the average of $f$ over the whole set ${\mathcal{S}}$. This phenomenon was first noticed by Panyushev in 2007 in the context of the rowmotion action on the set of antichains of a root poset; Armstrong, Stump, and Thomas proved Panyushev's conjecture in 2011. We describe a theoretical framework for results of this kind that applies more broadly, giving examples in a variety of contexts. These include linear actions on vector spaces, sandpile dynamics, Suter's action on certain subposets of Young's Lattice, Lyness 5-cycles, promotion of rectangular semi-standard Young tableaux, and the rowmotion and promotion actions on certain posets. We give a detailed description of the latter situation for products of two chains.


Introduction
We begin with the definition of our main unifying concept, and supporting nomenclature.
Definition 1. Given a set S, an invertible map τ from S to itself such that each τ -orbit is finite, and a function (or "statistic") f : S → K taking values in some field K of characteristic zero, we say the triple (S, τ, f ) exhibits homomesy iff there exists a constant c ∈ K such that for every τ -orbit O ⊂ S 1 #O x∈O f (x) = c. (1) In this situation we say that the function f : S → K is homomesic 1 under the action of τ on S, or more specifically c-mesic.
Typically our statistic f will be a natural measure (e.g., cardinality) and our set S will be a finite collection of combinatorial objects (e.g., order ideals in a poset), in which case homomesy can be restated equivalently as all orbit-averages being equal to the global average: We will also apply the term homomesy more broadly to include the case that the statistic f takes values in a vector space over a field of characteristic 0 (as in sections 2.4 and 2.6).
We have found many instances of (2) where the actions τ and the statistics f are natural ones. Many (but far from all) situations that support examples of homomesy also support examples of the cyclic sieving phenomenon of Reiner, Stanton, and White [RSW04] and more exploration of the linkages and differences is certainly in order. At the stated level of generality this notion appears to be new, but specific instances can be found in earlier literature. In particular, in 2007, Panyushev [Pan08] conjectured and in 2011, Amstrong, Stump, and Thomas [AST11] proved that if S is the set of antichains in the root poset of a finite Weyl group, Φ is the operation variously called the Brouwer-Schrijver map [BS74], the Fon-der-Flaass map [Fon93,CF95], the reverse map [Pan08], Panyushev complementation [AST11], and rowmotion [SW12], and f (A) is the cardinality of the antichain A, then (S, Φ, f ) satisfies (2).
Examples of homomesy are given starting with Section 2. These include the following triples (S, τ, f ), each of which will be explained in more detail in what follows. 7. Suter's map on Young diagrams with a weighted cardinality statistic.
Our main results for this paper involve studying the action of this rowmotion operator and also the (Striker-Williams) promotion operator on the poset P = [a] × [b], the product of two chains. (Here and throughout this article we use [n] to denote the set {1, 2, . . . , n} and the associated n-element poset.) We show that the statistic f := #A, the size of the antichain, is homomesic with respect to the promotion operator, and that the statistic f = #I(A), the size of the corresponding order ideal, is homomesic with respect to both the promotion and rowmotion operators.
Although these results are of intrinsic interest, we think the main contribution of the paper is its identification of homomesy as a phenomenon that (as we expect future articles to show) occurs quite widely. Within any linear space of functions on S, the functions that are homomesic under τ , like the functions that are invariant under τ , form a subspace. There is a loose sense in which the notions of invariance and homomesy (or, more strictly speaking, 0-mesy) are complementary; an extremely clean case of this complementarity is outlined in subsection 2.4, and a related complementarity (in the context of continuous rather than discrete orbits) is sketched in subsection 2.5. This article gives a general overview of the broader picture as well as a few specific examples done in more detail for the operators of promotion and rowmotion on the poset The authors are grateful to Drew Armstrong, Anders Björner, Joyce Chu, Barry Cipra, Karen Edwards, Robert Edwards, Darij Grinberg, Shahrzad Haddadan, Greg Kuperberg, Svante Linusson, Vic Reiner, Ralf Schiffler, Richard Stanley, Jessica Striker, Peter Winkler and Ben Young for useful conversations. Mike LaCroix wrote fantastic postscript code to generate animations and pictures that illustrate our maps operating on order ideals on products of chains. Ben Young also provided a diagram which we modified for Figure 6. An anonymous referee made very helpful suggestions for improving the "extended abstract" version of this paper that was presented at the 25th annual conference on Formal Power Series and Algebraic Combinatorics, held in Paris in June 2013. Another anonymous referee provided stimulating ideas as well as very helpful recommendations for improving the exposition. Several of our ideas were first incubated at meetings of the long-running Cambridge Combinatorial Coffee Club (CCCC), organized by Richard Stanley.

Examples of Homomesy
Here we give a variety of examples of homomesy in combinatorics, the first two of which long predate the general notion of homomesy; we also give non-combinatorial examples that establish links with different branches of mathematics. For examples of homomesy associated with piecewise-linear maps and birational maps, see [EP13].

Ballot theorems
Let S be the set of strings (s 1 , s 2 , . . . , s n ) of length n = a + b, consisting of a terms equal to −1 and b terms equal to +1, with a, b ≥ 1; we think of each such string as an order for counting n ballots in a two-way election, a of which are for candidate A and b of which are for candidate B. If a < b, then candidate B will be deemed the winner once all a + b ballots have been counted, and we ask for the probability that at every stage in the counting of the ballots, candidate B is in the lead. This probability is the same as the expected value of f (s), where f (s) is 1 if s 1 + · · · + s i > 0 for all 1 ≤ i ≤ n and is 0 otherwise, and where s is chosen uniformly at random from S. Bertrand's Theorem states that this probability is We can recast the famous "cycle lemma" proof of Bertrand's Theorem, due to Dvoretzky and Motzkin [DM47], in our framework as follows: Let τ := C L : S → S be the leftward cyclic shift operator that sends (s 1 , s 2 , s 3 , . . . , s n ) to (s 2 , s 3 , . . . , s n , s 1 ). Then over any orbit O one has s∈O f (s) = b − a (see [R07] for details), implying Dvoretzky and Motzkin used this method to prove a more general version of Bertrand's Theorem due to Barbier [B1887]: if b > ra for some positive integer r, then the probability that throughout the counting candidate B always has more than r times as many votes as candidate A is (b − ra)/(a + b).

Inversions in two-element multiset permutations
Let S be the same set of words as in the previous subsection, with f (s) := inv(s) := #{i < j : s i > s j }. For fixed i < j, the number of s in S with s i > s j (i.e., with s i = 1 and s j = 0) is n−2 a−1 , so the probability that an s chosen uniformly at random from S satisfies s i < s j is n−2 a−1 / n a = ab n(n−1) , so by additivity of expectation, the expected value of inv(s) is In other words, the inversion statistic is homomesic under the action of cyclically rotating bitstrings.
One way to prove Proposition 2 is to write For all s, i<j 1 is n(n−1) 2 and i<j s i s j is Since the average value of i<j (s i − s j ) over each orbit is 0, the average value of f over each orbit is ab/2. Proposition 2 it also follows from one of our results from Section 3. More specifically, we have a bijection between order-ideals in the poset P = [a] × [b] and strings consisting of a −1's and b +1's. Then promotion on J(P ) is equivariant with leftward cyclic shift on strings, and the cardinality of an order ideal is equal to the number of inversions in the associated string. Theorem 14 then yields the claimed result.
In the specfic case a = b = 2, the six-element set S decomposes into two orbits, shown in Figure 1. (Here we recode the elements of S as ordinary bit-strings, representing +1 and −1 by 1 and 0, respectively.) As frequently happens, not all orbits are the same size; however, one could also view the orbit of size 2 as a multiset orbit of size 4, cycling through the same set of elements twice. This perspective, where we view all orbits as multiset orbits of the same size, facilitates the discussion of certain comparisons.

Linear actions on vector spaces
Let V be a (not necessarily finite-dimensional) vector space over a field K of characteristic zero, and define f (v) = v (that is, our "statistic" is just the identity function). Let T : V → V be a linear map such that T n = I (the identity map on V ) for some fixed n ≥ 1.
Every v ∈ V can be written uniquely as the sum of an invariant vector v and a 0-mesic vectorv. (One suggestive way of paraphrasing the above is: Every element of the kernel of I − T n = (I − T )(I + T + T 2 + · · · + T n−1 ) can be written uniquely as the sum of an element of the kernel of I − T and an element of the kernel of I + T + T 2 + · · · + T n−1 .) For, one can check that v = v +v is such a decomposition, with v = (v + T v + · · · + T n−1 v)/n and v = v − v, and no other such decomposition is possible because that would yield a nonzero vector that is both invariant and 0-mesic, which does not exist. In representation-theoretic terms, we are applying symmetrization to v to extract from it the invariant component v associated with the trivial representation of the cyclic group, and the homomesic (0-mesic) componentv consists of everything else. This picture relates more directly to our earlier definition if we use the dual space V * of linear functionals on V as the set of statistics on V . As a concrete example, let V = R n and let T be the cyclic shift of coordinates sending (x 1 , x 2 , ..., x n ) to (x n , x 1 , ..., x n−1 ).

A circle action
Let S be the set of (real-valued) functions f (t) satisfying the differential equation can be written as a polynomial in f and f . Given an element p of the ring R[x, y], we will = c for all f in S. For example, x 2 + y 2 is invariant (one can think of this quantity as the total energy of a harmonic oscillator, with invariance corresponding to conservation of energy) and the x and y are 0-mesic (one can think of these as the mean displacement and mean velocity of a harmonic oscillator). We can give a basis for R[x, y] (viewed as a vector space over R) consisting of the constant 1 and the powers of x 2 + y 2 (which jointly span the "invariant" subspace of R[x, y]) along with the functions x, y, xy, x 2 − y 2 , x 3 , x 2 y, xy 2 , y 3 , etc. (which jointly span the 0-mesic subspace of R[x, y]).
Proposition 3. Let V n be the (n + 1)-dimensional vector subspace of R[x, y] spanned by the monomials x a y b with a + b = n. When n is odd, all of V n is 0-mesic. When n is even, V n can be written as the direct sum of an n-dimensional subspace of homomesic functions and a 1-dimensional subspace of functions that are invariant under time-evolution.
Proof. Define S 1 p(x, y) = 1 2π 2π 0 p(cos t, sin t) dt. Consider the monomial x a y b with a + b = n. If a (resp. b) is odd, the involution (x, y) → (−x, y) (resp. (x, −y)) shows that S 1 x a y b = 0. If a and b are both even, then S 1 x a y b is some positive number c a,b . Now let a, b vary subject to a+b = n. If n is odd, then x a y b is 0-mesic for all a, b with a+b = n (since at least one of a, b is odd), so all of V n is 0-mesic. If n is even, then for a, b even, (1/c a,b )x a y b − (1/c n,0 )x n y 0 is 0mesic, and these functions span an (n/2)-dimensional space; adding in the 0-mesic functions x a y b with a, b odd we get an (n − 1)-dimensional space of homomesies, complementing the 1-dimensional space of invariants spanned by (x 2 + y 2 ) n/2 . Greg Kuperberg points out that another term one might give to the 0-mesic subspace of R[x, y] in this context is the coinvariant kernel, since this subspace is the mutual kernel of all invariant linear maps from R[x, y] to R. Here, as in the preceding section, we get a clean complementarity between invariance and homomesy. That is, every element in R[x, y] can be written uniquely as the sum of an invariant element and a 0-mesic element.

Sandpile dynamics
Let G be a finite directed graph with vertex set V . For v ∈ V let outdeg(v) be the number of directed edges emanating from v, and for v, w ∈ V let deg(v, w) be the number of directed edges from v to w (which we will permit to be larger than 1, even when v = w). Define the combinatorial Laplacian of G as the matrix ∆ (with rows and columns indexed by the vertices of V ) whose v, vth entry is outdeg(v) − deg(v, v) and whose v, wth entry for v = w is − deg(v, w). Specify a global sink t with the property that for all v ∈ V there is a forward path from v to t, let V − = V \ {t}, and let ∆ (the reduced Laplacian) be the matrix ∆ with the row and column associated with t removed. By the Matrix-Tree theorem, ∆ is nonsingular. A sandpile configuration on G (with sink at t) is a function σ from V − to the nonnegative integers. (For more background on sandpiles, see Holroyd, Levine, Mészáros, Peres, Propp, & Wilson [HLMPPW08].) We say σ is stable if σ(v) < outdeg(v) for all v ∈ V − . For any sandpile-configuration σ, Dhar's least-action principle for sandpile dynamics (see Levine & Propp [LP10]) tells us that the set of nonnegative-integer-valued functions u on V − such that σ − ∆ u is stable has a minimal element φ = φ(σ) in the natural (pointwise) ordering; we call φ the firing vector for σ and we call σ − ∆ φ the stabilization of σ, denoted by σ • . If we choose a source vertex s ∈ V − , then we can define an action on sandpile configurations via τ (σ) = (σ + 1 s ) • , where 1 v denotes the function that takes the value 1 at v and 0 elsewhere. Say that σ is recurrent (relative to s) if τ m (σ) = σ for some m > 0. (This notion of recurrence is slightly weaker than that of [HLMPPW08]; they are equivalent when every vertex is reachable by a path from s.) Then τ restricts to an invertible map from the set of recurrent sandpile configurations to itself. Let ; if we average this relation over all σ in a particular τ -orbit, the left side telescopes, giving 0 = 1 s − ∆ f , where f denotes the average of f over the orbit. Hence: Proposition 4. Under the action of τ on recurrent sandpile configurations described above, the function f : σ → φ(σ + 1 s ) is homomesic, and its orbit-average is the function f * on V − such that ∆ f * = 1 s (unique because ∆ is nonsingular).
(0, 1, 0) = 1 s : We should mention that in this situation all orbits are of the same cardinality. This is a consequence of the fact that the set of recurrent chip configurations can be given the structure of a finite group (the "sandpile group" of G). For, given any finite abelian group G and any element h ∈ G, the action of h on G by multiplication has orbits that are precisely the cosets of G/H, where H is the subgroup of G generated by h, and all these cosets have size |H|.
Similar instances of homomesy were known for a variant of sandpile dynamics called rotor-router dynamics; see Holroyd-Propp [HP10]. It was such instances of homomesy that led the second author to seek instances of the phenomenon in other, better-studied areas of combinatorics.

Suter's map
This subsection needs to be written.

Promotion and rowmotion in products of two chains
For a finite poset P , we let J(P ) denote the set of order ideals (or down-sets) of P , F (P ) denote the set of (order) filters (or up-sets) of P , and A(P ) be the set of antichains of P . (For standard definitions and notation about posets and ideals, see Stanley [S11].) There is a bijection J(P ) ↔ A(P ) given by taking the maximal elements of I ∈ J(P ) or conversely by taking the order ideal generated by an antichain A ∈ A(P ). Similarly, there is a bijection F (P ) ↔ A(P ). Composing these with the complementation bijection between J(P ) and F (P ) leads to an interesting map that has been studied in several contexts [BS74, Fon93, CF95, Pan08, AST11, SW12], namely Φ A := A(P ) → J(P ) → F (P ) → A(P ) and the companion map Φ J := J(P ) → F (P ) → A(P ) → J(P ), where the subscript indicates whether we consider the map to be operating on antichains or order ideals. We often drop the subscript and just write Φ when context makes clear which is meant. Following Striker and Williams [SW12] we call this map rowmotion.
Let [a] × [b] denote the poset that is a product of chains of lengths a and b. Figure 5 shows an orbit of the action of Φ J starting from the ideal generated by the antichain {(2, 1)}. Note that the elements of [4] × [2] here are represented by the squares rather than the points in the picture, with covering relations represented by shared edges. One can also view this as an orbit of Φ A if one just considers the maximal elements in each shaded order ideal.
This section contains our main specific results, namely that the following triples exhibit homomesy: Here ∂ J is the promotion operator to be defined in the next subsection, and #I (resp. #A) denotes the statistic on J(P ) (resp. A(P )) that is the cardinality of the order ideal I (resp. the antichain A). All maps operate on the left (e.g., we write ∂ J I, not I∂ J ).

Background on the toggle group
Several of our examples arise from the toggle group of a finite poset (first explicitly defined in [SW12]; see also [CF95,Sta09,SW12]). We review some basic facts and provide some pointers to relevant literature.
Corollary 9 ([SW12], Cor. 4.9). Let P be a graded poset of rank r, and set T k := |x|=k σ x , the product of all the toggles of elements of fixed rank k. (This is well-defined by Proposition 7.) Then the composition T 1 T 2 · · · T r coincides with Φ J , i.e., rowmotion is the same as toggling by ranks from top to bottom.
Definition 10. In this situation, we call the sets with constant j ranks (in accordance with standard poset terminology), sets with constant i files, sets with constant j − i positive fibers, and sets with constant j + i negative fibers. (The words "positive" and "negative" indicate the slopes of the lines on which the fibers lie in the Hasse diagram.) More specifically, the element (k, ) ∈ [a] × [b] belongs to rank k + − 2, to file − k, to positive fiber k, and to negative fiber .
To each order ideal I ∈ J([a] × [b]) we associate a lattice path of length a + b joining the points (−a, a) and (b, b) in the plane, where each step is of type (i, j) → (i + 1, j + 1) or of type (i, j) → (i + 1, j − 1), as follows. Given 1 ≤ k ≤ a and 1 ≤ ≤ b, represent (k, ) ∈ [a] × [b] by the square centered at ( − k, + k − 1) with vertices ( − k, + k − 2), ( − k, + k), ( − k − 1, + k − 1), and ( − k + 1, + k − 1). Then the squares representing the elements of the order ideal I form a "Russian-style" Young diagram whose upper border is a path joining some point on the line of slope −1 to some point on the line of slope +1. Adding extra edges of slope −1 at the left and extra edges of slope +1 at the right, we get a path joining (−a, a) to (b, b). See Figures 3, 4, and 5 for several examples of this correspondence.
Definition 11. We can think of this path as the graph of a (real) piecewise-linear function h I : [−a, b] → [0, a + b]; we call this function (or its restriction to [−a, b] ∩ Z) the height function representation of the ideal I. To this height function we can in turn associate a word consisting of a −1's and b +1's, whose ith term (for we call this the sign-word associated with the order ideal I. Note that the sign-word simply lists the slopes of the segments making up the path, and that either the sign-word or the height-function encodes all the information required to determine the order ideal. So to prove that the cardinality of I is homomesic, it suffices to prove that the function h I (−a) + h I (−a + 1) + · · · + h I (b) is homomesic (where our combinatorial dynamical system acts on height functions h via its action on order ideals I).

Promotion in products of two chains
In general a ranked poset P may not have an embedding in Z × Z that allows files to be defined; when they are, however, then all toggles corresponding to elements within the same file commute by Proposition 7, so their product is a well defined operation on J(P ). This allows one to define an operation on J(P ) by successively toggling all the files from left to right, in analogy to Corollary 9.  . This definition and their results apply more generally to the class they define of rc-posets, whose elements fit neatly into "rows" and "columns" (which we call here "ranks" and "files"). As with Φ, we can think of ∂ as operating either on J(P ) or A(P ), adding subscripts ∂ J or ∂ A if necessary. Since the cyclic left-shift has period a + b, so does ∂. Proof. To show that #I is homomesic, by Proposition 12 it suffices to show that h I (k) is homomesic for all −a ≤ k ≤ b. Note that here we are thinking of I as varying over J(P ), and h I as being a function-valued function on J(P ).
We can write h I (k) as the telescoping sum h I (−a) + (h I (−a + 1) − h I (−a)) + (h I (−a + 2) − h I (−a + 1)) + · · · + (h I (k) − h I (k − 1)); to show that h I (k) is homomesic for all k, it will be enough to show that all the increments h I (k) − h I (k − 1) are homomesic. Note that these increments are precisely the terms of the sign-word of I. Create a square array with a + b rows and a + b columns, where the rows are the sign-words of I and its successive images under the action of ∂; each row is just the cyclic left-shift of the row before. Since each row contains a −1's and b +1's, the same is true of each column. Thus, for all k, the average value The next example shows that the cardinality of the antichain A I associated with the order ideal I is not homomesic under the action of promotion ∂.

Rowmotion in products of two chains
Unlike promotion, the rowmotion operator turns out to exhibit homomesy with respect to both the statistic that counts the size of an order ideal and the statistic that counts the size of an antichain.

Rowmotion on order ideals in J([a] × [b])
We can describe rowmotion nicely in terms of the sign-word. We define blocks within the sign-word as occurrences of the subword −1, +1 (that is, a −1 followed immediately by a +1). Once we have found all the blocks, we identify all the gaps between the blocks, where a gap is bounded by two consecutive blocks, or between the beginning of the word  Create a square array with a+b rows and a+b columns, where the rows are the sign-words of I and its successive images under the action of Φ. Consider any two consecutive columns of the array, and the width-2 subarray they form. There are just four possible combinations of values in a row of the subarray: (+1, +1), (+1, −1), (−1, +1), and (−1, −1). However, we have just remarked that a row is of type (−1, +1) if and only if the next row is of type (+1, −1) (where we consider the row after the bottom row to be the top row). Hence the number of rows of type (−1, +1) equals the number of rows of type (+1, −1). It follows that any two consecutive column-sums of the full array are equal, since other row types contribute the same value to each column sum. That is, within the original square array, every two consecutive columns have the same column-sum. Hence all columns have the same columnsum. This common value of the column-sum must be 1/(a + b) times the grand total of the values of the square array. But since each row contains a −1's and b +1's, each row-sum is b − a, so the grand total is (a + b)(b − a), and each column-sum is b − a. Since this is independent of which rowmotion orbit we are in, we have proved homomesy for elements of the sign-word of I as I varies over J([a] × [b]), and this gives us the desired result about #I, just as in the proof of Theorem 14.

+1, if A has an element in row i (for i ∈ [a]) or
A has NO element in column i (for a + 1 ≤ i ≤ n); −1 otherwise.
Proposition 19 (Stanley-Thomas). The correspondence A ←→ w A is a bijection from A([a] × [b]) to binary words w ∈ {−1, +1} a+b with exactly a −1's and b +1's. Furthermore, this bijection is equivariant with respect to the actions of rowmotion Φ A and rightward cyclic shift C R .
Note that the classical result that Φ a+b A is the identity map follows immediately. Proof. It suffices to prove a more refined claim, namely, that if S is any row or column of [a] × [b], the cardinality of A ∩ S is homomesic under the action of rowmotion on A. By the previous result, rowmotion corresponds to cyclic shift of the Stanley-Thomas word, and the entries in the Stanley-Thomas word tell us which fibers (rows or columns) contain an element of A and which do not. Specifically, for 1 ≤ k ≤ a, if S is the kth row, then A intersects S iff the kth symbol of the Stanley-Thomas word is a +1. Since the Stanley-Thomas word contains a −1's and b +1's, the multiset orbit of A of size a + b has exactly b elements that are antichains that intersect S. That is, the sum of #(A ∩ S) over the multiset orbit of size a + b is exactly b, for each of the a rows of [a] × [b]. Summing over all the rows, we see that the sum of #A over the multiset orbit is ab. Hence #A is homomesic with average ab/(a + b).