On flow polytopes, order polytopes, and certain faces of the alternating sign matrix polytope
In this paper we study an alternating sign matrix analogue of the Chan-Robbins-Yuen polytope, which we call the ASM-CRY polytope. We show that this polytope has Catalan many vertices and its volume is equal to the number of standard Young tableaux of staircase shape; we also determine its Ehrhart polynomial. We achieve the previous by proving that the members of a family of faces of the alternating sign matrix polytope which includes ASM-CRY are both order and flow polytopes. Inspired by the above results, we relate three established triangulations of order and flow polytopes, namely Stanley’s triangulation of order polytopes, the Postnikov-Stanley triangulation of flow polytopes and the Danilov-Karzanov-Koshevoy triangulation of flow polytopes. We show that when a graph is a planar graph, in which case the flow polytope is also an order polytope, Stanley’s triangulation of this order polytope is one of the Danilov-Karzanov-Koshevoy triangulations of . Moreover, for a general graph we show that the set of Danilov-Karzanov-Koshevoy triangulations of is a subset of the set of Postnikov-Stanley triangulations of . We also describe explicit bijections between the combinatorial objects labeling the simplices in the above triangulations.
In this paper, we study a family of faces of the alternating sign matrix polytope inspired by an intriguing face of the Birkhoff polytope: the Chan-Robbins-Yuen (CRY) polytope . We call this family of faces the ASM-CRY family of polytopes. Interest in the CRY polytope centers around its volume formula as a product of consecutive Catalan numbers; this has been proved , but the problem of finding a combinatorial proof remains open. We prove that the polytopes in the ASM-CRY family are order polytopes and use Stanley’s theory of order polytopes  to give a combinatorial proof of formulas for their volumes and Ehrhart polynomials. We also show that these polytopes, and all order polytopes of strongly planar posets, are flow polytopes. This observation brings us to the general question of relating the different known triangulations of flow and order polytopes. We show that when is a planar graph, in which case the flow polytope of is also an order polytope, then Stanley’s canonical triangulation of this order polytope  is one of the Danilov-Karzanov-Koshevoy triangulations of the flow polytope of , a statement first observed by Postnikov . Moreover, for general we show that the set of Danilov-Karzanov-Koshevoy triangulations of the flow polytope of is a subset of the set of framed Postnikov-Stanley triangulations of the flow polytope of [15, 19]. We also describe explicit bijections between the combinatorial objects labeling the simplices in the above triangulations, answering a question posed by Postnikov .
We highlight the main results of the paper in the following theorems. While we define some of the notation here, some only appears in later sections to which we give pointers after the relevant statements.
In Definition 4.1, we define the ASM-CRY family of polytopes indexed by partitions where . In Theorem 4.3, we prove that the polytopes in this family are faces of the alternating sign matrix polytope defined in [4, 22]. In the case when we obtain an analogue of the Chan-Robbins-Yuen (CRY) polytope, which we call the ASM-CRY polytope, denoted by . Our main theorem about this family of polytopes is the following. For the necessary definitions, see Sections 3.3 and 4.
The polytopes in the family are affinely equivalent to flow and order polytopes. In particular, is affinely equivalent to the order polytope of the poset and the flow polytope .
By Stanley’s theory of order polytopes  it follows that the volume of the polytope for any is given by the number of linear extensions of the poset (the number of Standard Young Tableaux of skew shape , and its Ehrhart polynomial is given by the order polynomial of the poset (counting weak plane partitions of skew shape with bounded parts). See Corollary 4.7 for the general statement. We give the application to in the corollary below. For further examples of polytopes in , see Figure 7.
is affinely equivalent to the order polytope of the poset . Thus, has vertices, its normalized volume is given by
and its Ehrhart polynomial is
In Theorems 3.8 and 3.10, we make explicit the relationship between flow and order polytopes, showing that they correspond under certain planarity conditions. For an introduction to flow and order polytopes, see Section 3, and for the definitions of and , see Definition 4.4 and the discussion before Theorem 3.10, respectively.
As mentioned earlier, a canonical triangulation of order polytopes was given by Stanley , and two families of triangulations of flow polytopes were constructed by Postnikov and Stanley [15, 19] as well as Danilov, Karzanov and Koshevoy . It is natural to try to understand the relation among these triangulations, and we prove the following results, the first of which was first observed by Postnikov . For the necessary definitions, see Sections 5 and 6.
Theorem 1.3 (Postnikov ).
Given a planar graph , the canonical triangulation of the order polytope is equal to the Danilov-Karzanov-Koshevoy triangulation of the flow polytope coming from the planar framing.
Given a framed graph , the set of Danilov-Karzanov-Koshevoy triangulations of the flow polytope is a subset of the set of framed Postnikov-Stanley triangulations of .
The outline of the paper is as follows. In Section 2, we discuss the Birkhoff and alternating sign matrix polytopes, as well as some of their faces. In Section 3, we give background information on flow and order polytopes and show that flow polytopes of planar graphs are order polytopes and that order polytopes of strongly planar posets are flow polytopes. In Section 4, we study a family of faces of the alternating sign matrix polytopes and show that they are affinely equivalent to both flow and order polytopes and calculate their volumes and Ehrhart polynomials in particularly nice cases. In Section 5, we study triangulations of flow polytopes of planar graphs (which include the polytopes of Section 4) and show that their canonical triangulations defined by Stanley  are also Danilov-Karzanov-Koshevoy triangulations . Finally, in Section 6, we study triangulations of flow polytopes of an arbitrary graph, that is, the Danilov-Karzanov-Koshevoy triangulations and the framed Postnikov-Stanley triangulations. We show that the former is a subset of the latter. We also exhibit explicit bijections between the combinatorial objects indexing the various triangulations, answering a question raised by Postnikov .
2. Faces of the Birkhoff and alternating sign matrix polytopes
In this section, we explain the motivation for our study of certain faces of the alternating sign matrix polytope. If is an integral polytope, its Ehrhart polynomial is the polynomial that counts the number of lattice points of the dilated polytope . In this case the relative volume of is the leading term of and its normalized volume is the product of its relative volume and . We start by defining the Birkhoff and Chan-Robbins-Yuen polytopes; we then define the alternating sign matrix counterparts.
The Birkhoff polytope, , is defined as
Matrices in are called doubly-stochastic matrices. A well-known theorem of Birkhoff  and Von Neumann  states that , as defined above, equals the convex hull of the permutation matrices. Note that has facets and dimension , its vertices are the permutation matrices, and its volume has been calculated up to by Beck and Pixton . De Loera, Liu and Yoshida  gave a closed summation formula for the volume of , which, while of interest on its own right, does not lend itself to easy computation. Shortly after, Canfield and McKay  gave an asymptotic formula for the volume.
A special face of the Birkhoff polytope, first studied by Chan-Robbins-Yuen , is as follows.
The Chan-Robbins-Yuen polytope, , is defined as
Theorem 2.3 (Zeilberger ).
The proof in  used a relation (see Theorem 3.3) expressing the volume as a value of the Kostant partition function (see Definition 3.4) and a reformulation of the Morris constant term identity  to calculate this value. No combinatorial proof is known.
Next we give an analogue of the Birkhoff polytope in terms of alternating sign matrices. Recall that alternating sign matrices (ASMs)  are square matrices with the following properties:
the entries in each row/column sum to 1, and
the nonzero entries along each row/column alternate in sign.
The ASMs with no negative entries are the permutation matrices. See Figure 1 for an example.
The alternating sign matrix polytope, , is defined as follows:
where we have the first sum for any , the second sum for any , the third sum for any , and the fourth sum for any .
Behrend and Knight , and independently Striker , defined . The alternating sign matrix polytope can be seen as an analogue of the Birkhoff polytope, since the former is the convex hull of all alternating sign matrices (which include all permutation matrices) while the latter is the convex hull of all permutation matrices. The polytope has facets (for ) , its dimension is , and its vertices are the alternating sign matrices [4, 22]. The Ehrhart polynomial has been calculated up to . Its normalized volume for is calculated to be
and no asymptotic formula for its volume is known.
In analogy with , we study a special face of the ASM polytope we call the ASM-CRY polytope (and show, in Theorem 4.3, it is indeed a face of ).
The ASM-CRY polytope is defined as follows.
Since the polytope has a nice product formula for its normalized volume, it is then natural to wonder if the volume of the alternating sign matrix analogue of , which we denote by , is similarly nice. In Theorem 1.1 and Corollary 1.2, we show that is both a flow and order polytope, and using the theory established for the latter, we give the volume formula and the Ehrhart polynomial of . Just like in the case, all formulas obtained are combinatorial. Unlike in the case, all the proofs involved are combinatorial. In Theorem 1.1, we extend these results to a family of faces of the ASM polytope, of which is a member; see Section 4.
3. Flow and order polytopes
In order to state and prove Theorem 1.1 in Section 4, we need to discuss flow and order polytopes. In Section 3.1, we define flow and order polytopes and also explain how to see as the flow polytope of the complete graph. In Sections 3.2 and 3.3, we prove that the flow polytope of a planar graph is the order polytope of a related poset, and vice versa.
3.1. Background and definitions
Let be a connected graph on the vertex set with edges directed from the smallest to largest vertex. We assume that each vertex has both incoming and outgoing edges. Denote by the smallest (initial) vertex of edge and the biggest (final) vertex of edge .
A flow of size one on is a function such that
The flow polytope associated to the graph is the set of all flows of size one on .
Note that the restriction that at each vertex of there are both incoming and outgoing edges is not a serious one. If there is a vertex with only incoming or outgoing edges, then in the flow on all these edges must be zero, and thus, up to removing such vertices, any flow polytope is equivalent to a flow polytope defined as above.
The polytope is a convex polytope in the Euclidean space and its dimension is (e.g. see ). The vertices of are given by unit flows along maximal directed paths or routes of from the source () to the sink () [17, §13]. Figure 3 shows the equations of and explains why this polytope is equivalent to . The same correspondence shows that and coincide. The following theorem connects volumes of flow polytopes and Kostant partition functions.
For a loopless graph on the vertex set , with ,
where is the Kostant partition function and is normalized volume.
Recall the definition of the Kostant partition function.
The Kostant partition function is the number of ways to write the vector as a nonnegative linear combination of the positive type roots corresponding to the edges of , without regard to order. The edge , , of corresponds to the vector , where is the standard basis vector in .
It is easy to see by definition that the Ehrhart polynomial of in variable is equal to
Now we are ready to define order polytopes and relate them to flow polytopes.
Definition 3.5 (Stanley ).
The order polytope, , of a poset with elements is the set of points in with and if then . We identify each point of with the function with .
In general, computing or finding a combinatorial interpretation for the volume of a polytope is a hard problem. Order polytopes are an especially nice class of polytopes whose volume has a combinatorial interpretation.
Theorem 3.6 (Stanley ).
Given a poset we have that
the vertices of are in bijection with the order ideals of ,
the normalized volume of is , where is the number of linear extensions of ,
the Ehrhart polynomial of equals the order polynomial of .
Given a poset and a positive integer , the order polynomial is the number of order preserving maps .
3.2. Flow polytopes of planar graphs are order polytopes
The following theorem, which states that a flow polytope of a planar graph is an order polytope, is a result communicated to us by Postnikov . We need the following conventions. Given a connected graph on the vertex set , we draw it in the plane so that the vertices are on a horizontal line in this order. We say that is planar if it has a planar embedding with fixed on a horizontal line. See Figure 4. Given such a planar embedding of , we draw the truncated dual graph of , denoted , which is the dual graph with the vertex corresponding to the infinite region deleted together with its incident edges. Note that since the vertices are drawn on a horizontal line, we can naturally orient the edges of from “lower” to “higher” (see Figure 4 (b)). The poset is then obtained by considering as its Hasse diagram (see Figure 4 (c)). Note that by Euler’s formula, which equals .
Theorem 3.8 (Postnikov ).
Let be a planar graph on the vertex set such that at each vertex there are both incoming and outgoing edges. Fix a planar drawing of . Then there is a linear map from the flow polytope to the order polytope which preserves relative volume.
For an element of let , where the sum is taken over the edges that are intersected by a fixed path from the element to the “low point” in the dual graph of . The “low point” of the dual graph is the vertex corresponding to the infinite face of and we draw it below the graph as shown on Figure 5 (a). It is easy to see that due to flow conservation this map is well-defined, that is it does not depend on the path we choose.
In addition, we have that since for all edges . Also, since the set of edges whose sum of flows equals can always be extended to a cut of the graph whose flow is the total flow present in the graph. Next, if covers in then there is an edge in separating the graph faces and . Thus . Hence the linear map mentioned in the theorem takes a point of to the point of the order polytope . This map preserves the integer points in the affine spans of these polytopes, thereby preserving relative volume. ∎
By Theorem 3.8, if is a planar graph then is equivalent to an order polytope. This raises the question of whether this relation holds for non-planar graphs: for instance for the polytope for . We can use a similar construction to that in Theorem 3.8 to show that and are equivalent to the order polytopes of the posets:
We leave it as a question whether (dimension , vertices, volume ) is an order polytope.
3.3. Order polytopes of strongly planar posets are flow polytopes
We now give a converse of Theorem 3.8, showing that the order polytope of a strongly planar poset is a flow polytope. A poset is strongly planar if the Hasse diagram of has a planar embedding with all edges directed upward in the plane. Define the graph from by taking the Hasse diagram of and drawing in two additional edges from to , one which goes to the left of all the poset elements and another to the right. We can then define the graph to be the truncated dual of , provided that is planar. will be planar whenever is planar, which in turn is when is strongly planar. The orientation of is inherited from the poset in the following way: if in the construction of the truncated dual, the edge of crosses the edge where in , then is on the left and is on the right as you traverse .
If is a strongly planar poset, then there is a linear map from the order polytope to the flow polytope which preserves relative volume.
For an edge in , let , where in the dual construction, crosses the Hasse diagram edge . It is easy to see that this map is well-defined and it maps to by mapping to , where is as prescribed above. This map preserves the integer points in the affine spans of these polytopes, thereby preserving relative volume. ∎
4. and the family of polytopes
In this section, we introduce the ASM-CRY family of polytopes , which includes , and show that each of these polytopes is a face of the ASM polytope. We, furthermore, show that each polytope in this family is both an order and a flow polytope. Then, using the theory of order and flow polytopes as discussed in Section 3.1, we write down their volumes and Ehrhart polynomials.
Let be the staircase partition considered as the positions of an matrix given by . Let the partition denote matrix positions .
We define the ASM-CRY family
Note that .
In the following proposition we give a convex hull description of the polytopes in this family.
The polytope is the convex hull of the alternating sign matrices with .
Let denote the convex hull of the alternating sign matrices with . It is easy to see that is contained in , since matrices in both polytopes have the same prescribed zeros and satisfy the inequality description of the full ASM polytope .
It remains to prove that is contained in . Suppose there exists a matrix such that . We know that is in the convex hull of all ASMs. So , where are distinct alternating sign matrices and . At least one of these ASMs, say must have a nonzero entry for some satisfying either . Suppose ; the argument follows similarly in the case . Now since and , there must be another ASM, say such that is nonzero of opposite sign. Say and . Then by the definition of an alternating sign matrix, there must be such that . But as well, so there must be an with and such that . Eventually, we will reach the border of the matrix and reach a contradiction. Thus, . ∎
We show in Theorem 4.3 below that the polytopes in are faces of . First, we need some terminology from . Consider vertices on a square grid: ‘internal’ vertices and ‘boundary’ vertices , , , and , where . Fix the orientation of this grid so that the first coordinate increases from top to bottom and the second coordinate increases from left to right, as in a matrix. The complete flow grid is defined as the directed graph on these vertices with directed edges pointing in both directions between neighboring internal vertices within the grid, and also directed edges from internal vertices to neighboring border vertices. That is, has edge set . A simple flow grid of order is a subgraph of consisting of all the vertices of , and in which four edges are incident to each internal vertex: either four edges directed inward, four edges directed outward, or two horizontal edges pointing in the same direction and two vertical edges pointing in the same direction. An elementary flow grid is a subgraph of whose edge set is the union of the edge sets of simple flow grids. See Figure 6.
The polytope is a face of , of dimension . In particular, is a face of , of dimension .
In Proposition 4.2 of , it was shown that the simple flow grids of order are in bijection with the alternating sign matrices. In this bijection, the internal vertices of the simple flow grid correspond to the ASM entries; the sources correspond to the ones of the ASM, the sinks correspond to the negative ones, and all other vertex configurations correspond to zeros. In Theorem 4.3 of , it was shown that the faces of are in bijection with elementary flow grids, with the complete flow grid in bijection with the full ASM polytope . This bijection was given by noting that the convex hull of the ASMs in bijection with all the simple flow grids contained in an elementary flow grid is, in fact, an intersection of facets of the ASM polytope , and is thus a face of . Since, by Proposition 4.2, equals the convex hull of the ASMs in it, we need only show there exists an elementary flow grid whose contained simple flow grids correspond exactly to these ASMs.
We can give this elementary flow grid explicitly. We claim that the edge set where
is the union of the edge sets all the simple flow grids in bijection with ASMs in , thus the digraph with this edge set is an elementary flow grid. Furthermore, no other simple flow grid can be constructed from directed edges in this set, since such a simple flow grid would have to include an edge pointing in the wrong direction in either the region or . Thus, is a face of .
To calculate the dimension of , we use the following notion from . A doubly directed region of an elementary flow grid is a connected collection of cells in the grid completely bounded by double directed edges but containing no double directed edges in the interior. Theorem 4.5 of  states that the dimension of a face of equals the number of doubly directed regions in the corresponding elementary flow grid. The number of doubly directed regions in the elementary flow grid corresponding to equals . See Figure 6. ∎
Let and be as in Definition 4.1. Let be the poset with elements corresponding to the positions with partial order if and .
The polytopes in the family are affinely equivalent to flow and order polytopes. In particular, is affinely equivalent to the order polytope of the poset and the flow polytope .
We prove Theorem 1.1 by first using two lemmas to show that is affinely equivalent to the order polytope of the poset . Then since this poset is planar, by Theorem 3.10 its order polytope is affinely equivalent to the flow polytope .
Given a matrix , define the corner sum matrix by
For , let be the set of functions . We view the order polytope of as a subset of . Define by where . See the second map in Figure 8.
The image of is in , i.e. if then .
We first show that for all and . By the defining inequalities of the ASM polytope (see Definition 2.4), we have that the partial column sums satisfy for any and . So since is a sum of partial column sums, then as desired.
Next we show that for all . Note that it is not true that for all matrices in (for example the permutation matrix of has ). But we show for all as follows. The forced zeros and the requirement that each column sums to one imply that , since the rest of the column entries equal zero. Thus we also have for any . If then , since each and .
Now let . Note that the sum of the first rows satisfies . Also, by the discussion of the previous paragraph,
since . Finally,
since this is a sum of partial column sums. So we have
Thus so that .
The image of is in the order polytope .
By Lemma 4.5 we know that the image of is in . Note that if and , then , thus we have that if and only if in . So is in the order polytope . ∎
Proof of Theorem 1.1.
By Lemmas 4.5 and 4.6 we have that the map is an affine map from to with homogeneous part where is a -upper unitriangular matrix. Thus is volume preserving () and when restricted to a lattice is a bijection between the lattice points of and , . Thus, is a bijective affine map from to , showing that they are affinely equivalent (and thus combinatorially equivalent).
Finally since the poset is planar, by Theorem 3.10 is also affinely equivalent to the flow polytope . ∎
Corollary 4.7 ().
For in we have that its normalized volume is
and its Ehrhart polynomial is
Note that using Theorem 3.3 and the discussion below it, we can express the volume and Ehrhart polynomial of any flow polytope as a Kostant partition function. Thus, Theorem 1.1 gives us several Kostant partition function identities. In particular, Corollaries 1.2, 4.9 and 4.8 compute the volumes and Ehrhart polynomials of three subfamilies of polytopes in that are associated to posets with a nice number of linear extensions and vertices. This includes the ASM-CRY polytope. See Figure 7.
is affinely equivalent to the order polytope of the poset and the flow polytope . Thus, has vertices, its normalized volume is given by
and its Ehrhart polynomial is
When , is isomorphic to the order polytope of the poset (that is, the type positive root lattice). The number of linear extension of this poset is the number of standard Young tableaux (SYT) of shape ,
By Stanley’s theory of order polytopes (see Theorem 3.6) . When is an integer, counts the the number of plane partitions of shape with largest part . By an unpublished result of Proctor  (see also ) this number is given by the product formula in the RHS of (4.1). ∎
We give a few other examples of polytopes in the family that have known nice formulas for the volume, namely, in the cases for . See Figure 7.
Let be the poset with elements and no relations and denote the zigzag poset with elements: .
is isomorphic to the order polytope of . has vertices and its normalized volume equals .
Since the poset is an antichain, there are no relations, so the number of order ideals is and the number of linear extensions is . Thus, the result follows from Theorem 1.1. ∎
is affinely equivalent to the order polytope of . has number of vertices given by the Fibonacci number and normalized volume given by the Euler number .
The number of order ideals of is given by the Fibonacci number . The number of linear extensions of this poset is the number of SYT of skew shape which is given by the Euler number . Thus, the result follows from Theorem 1.1. ∎
For the case , the polytope is isomorphic to the order polytope of the poset . The number of vertices of the polytope (order ideals of the poset) is given by the number of Dyck paths with height at most [18, A211216], [11, §3.1]. The volume of the polytope is given by the number of skew SYT of shape . There are formulas for this number of SYT as determinants of Euler numbers (e..g see Baryshnikov-Romik ).
We now turn from our investigation of the family of polytopes to triangulations of flow and order polytopes.
5. Triangulations of flow polytopes of planar graphs
As we have seen in Section 3, flow polytopes of planar graphs are also order polytopes. Stanley  gave a canonical way of triangulating an order polytope . For a linear extension of the poset on elements , define the simplex
Note that the vertices of this simplex are of the form for . The simplices corresponding to all linear extensions of are top dimensional simplices in a triangulation of , which we refer to as the canonical triangulation of . There are also two known combinatorial ways of triangulating flow polytopes: one given by Postnikov and Stanley [15, 19], and one by Danilov, Karzanov and Koshevoy . In this section, we show that given a planar graph , the canonical triangulation of is also a triangulation obtained by the Danilov-Karzanov-Koshevoy method for . This result was first observed by Postnikov . We also construct a direct bijection between linear extensions of and maximal cliques of , which index the Danilov-Karzanov-Koshevoy triangulation of . In Section 6, we will prove for a general graph that the Danilov-Karzanov-Koshevoy triangulations of can also be obtained by a framed Postnikov-Stanley method. Thus, in particular, the canonical triangulation of for a planar graph is also in the set of the framed Postnikov-Stanley triangulations of .
5.1. The canonical triangulation of is a Danilov-Karzanov-Koshevoy triangulation of
Given a connected graph on the vertex set with edges oriented from smaller to bigger vertices, the vertices of the flow polytope correspond to integer flows of size one on maximal directed paths from the source () to the sink (). Following  we call such maximal paths routes. The following definitions also follow . Let be an inner vertex of whenever is neither a source nor a sink. Fix a framing at each inner vertex , that is, a linear ordering on the set of incoming edges to and the linear ordering on the set of outgoing edges from . A framed graph is a graph with a framing at each inner vertex. For a framed graph and an inner vertex , we denote by and by the set of maximal paths ending in and the set of maximal paths starting at , respectively. We define the order on the paths in as follows. If , then let be the largest vertex after which and