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 [CRY]. 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 [Z], 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 [Stop] 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 [Stop] is one of the Danilov-Karzanov-Koshevoy triangulations of the flow polytope of [kosh], a statement first observed by Postnikov [P13]. 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 [P13, S]. We also describe explicit bijections between the combinatorial objects labeling the simplices in the above triangulations, answering a question posed by Postnikov [P13].
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 LABEL:faces, we define the ASM-CRY family of polytopes indexed by partitions where . In Theorem LABEL:prop:asmfaces, we prove that the polytopes in this family are faces of the alternating sign matrix polytope defined in [BK, StrASM]. 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 LABEL:sec:cryasm.
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 [Stop] 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 LABEL:cor:ASMcor for the general statement. We give the application to in the corollary below. For further examples of polytopes in , see Figure LABEL:fig:family.
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 8 and 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 LABEL:star and the discussion before Theorem 10, respectively.
As mentioned earlier, a canonical triangulation of order polytopes was given by Stanley [Stop], and two families of triangulations of flow polytopes were constructed by Postnikov and Stanley [P13, S] as well as Danilov, Karzanov and Koshevoy [kosh]. 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 [P13]. For the necessary definitions, see Sections LABEL:sec:planar and LABEL:sec:tri.
Theorem 3 (Postnikov [P13]).
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 .
All three of the above-mentioned triangulations are indexed by natural sets of combinatorial objects and we give explicit bijections between these sets in Sections LABEL:sec:planar and LABEL:sec:tri.
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 LABEL:sec:cryasm, 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 LABEL:sec:planar, we study triangulations of flow polytopes of planar graphs (which include the polytopes of Section LABEL:sec:cryasm) and show that their canonical triangulations defined by Stanley [Stop] are also Danilov-Karzanov-Koshevoy triangulations [kosh]. Finally, in Section LABEL:sec:tri, 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 [P13].
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 [Birkhoff] and Von Neumann [VonNeumann] 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 [BP]. De Loera, Liu and Yoshida [LLY] 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 [CM] gave an asymptotic formula for the volume.
A special face of the Birkhoff polytope, first studied by Chan-Robbins-Yuen [CRY], is as follows.
The Chan-Robbins-Yuen polytope, , is defined as
has dimension and vertices. This polytope was introduced by Chan-Robbins-Yuen [CRY] and in [Z] Zeilberger calculated its normalized volume as the following product of Catalan numbers.
Theorem 3 (Zeilberger [Z]).
The proof in [Z] used a relation (see Theorem 3) expressing the volume as a value of the Kostant partition function (see Definition 4) and a reformulation of the Morris constant term identity [WM] 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) [ASM] 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.
Definition 4 (Behrend-Knight [Bk], Striker [StrASM]).
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 [BK], and independently Striker [StrASM], 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 ) [StrASM], its dimension is , and its vertices are the alternating sign matrices [BK, StrASM]. The Ehrhart polynomial has been calculated up to [BK]. 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 LABEL:prop:asmfaces, 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 and Corollary 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, we extend these results to a family of faces of the ASM polytope, of which is a member; see Section LABEL:sec:cryasm.
3. Flow and order polytopes
In order to state and prove Theorem 1 in Section LABEL:sec:cryasm, 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 [BV2]). The vertices of are given by unit flows along maximal directed paths or routes of from the source () to the sink () [schrijver, §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.
Theorem 3 (Postnikov-Stanley [P13, S], Baldoni-Vergne [Bv2]).
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 5 (Stanley [Stop]).
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 6 (Stanley [Stop]).
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 [P13]. 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 8 (Postnikov [P13]).
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 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 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 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.