1.1 Measuring combinatorial polytopes

Measuring is one of the central questions in mathematics: How do we quantify the size or complexity of a mathematical object? In the theory of polytopes, it is natural to measure a shape by means of its volume or its surface area. Computing these quantities for a highdimensional polytope P is a difficult task ^{Bárány & Füredi (1987)}; ^{Dyer & Frieze (1988)}, and one approach has been to discretize the question. One places the polytope P on a grid and asks: How many grid points does P contain? How many grid points do its dilates 2P, 3P, 4P, ... contain? This approach is illustrated in figure 1 for four polygons.

^{Ehrhart (1962)} showed that when the polytope P has integer (or rational) vertices, then there is a polynomial (or quasipolynomial) ehr _P (x) such that the dilate tP contains exactly ehr _P (t) grid points for any positive integer t. He also showed that the leading coefficient of ehr _P (x) equals the (suitably normalized) volume of P, and the second leading coefficient equals half of the (suitably normalized) surface area. Therefore the Ehrhart (quasi)polynomial (which we will define in detail in Section 2.1 below) is a more precise measure of size than these two quantities. Ehrhart theory is devoted to measuring polytopes in this way, computing continuous quantities discretely (see, e.g., ^{Fukuda (2008}. Electronically available at http://www.ifor.math.ethz.ch/∼fukuda/cddhome/cdd.html)).

Combinatorics studies the possibilities of a discrete situation; for example, the possible ways of reordering, or permuting the numbers 1,..., n. In most situations of interest, the number of possibilities of a discrete problem is tremendously large, so one needs to find intelligent ways of organizing them. Geometric combinatorics offers an approach: model the (discrete) possibilities of a problem with a (continuous) polytope. A classic example is the permutahedron Π _n , a polytope whose vertices are the n! permutations of {1, 2,..., n}. (Figure 2 shows the permutahedron Π₄.) One can answer many questions about permutations using the geometry of this polytope. In this way, the general strategy of geometric combinatorics is to model discrete problems continuously.

Combining these two forms of interplay between the discrete and the continuous, it is natural to begin with a discrete problem, model it in terms of a continuous polytope, and then measure that polytope discretely. ^{Stanley (1991)} pioneered this line of inquiry, with the following beautiful theorem.

Theorem 1.1 ( ^{Stanley (1991)}). The Ehrhart polynomial of the permutahedron Π _n is

where a _i is the number of graphs with i edges on the vertices {1,..., n} that contain no cycles. In particular, the normalized volume of the permutahedron Π _n is the number of trees on {1,..., n}, which equals n ⁿ⁻² .

1.2 Our results: measuring classical Coxeter permutahedra

The permutahedron Π _n is one of an important family of highly symmetric polytopes: the reduced, crystallographic standard Coxeter permutahedra; see Section 2.3 for a precise definition and some Lie theoretic context. These polytopes come in four infinite families A _n−1 , B _n ,C _n , D _n (n≥1) called the classical types, and five exceptions E ₆ , E ₇ , E ₈ , F ₄ , and G ₂. The standard Coxeter permutahedra of the classical types are the following polytopes in Rn:

Here a signed permutation of a sequence S is obtained from a permutation of S by introducing signs to the entries arbitrarily; the evenly signed permutations are those that introduce an even number of minus signs. Figure 1 shows the standard Coxeter permutahedra Π(A ₂), Π(B ₂), Π(C ₂), and Π(D ₂), as well as their second and third dilates. Note that the evenly signed permutations of {0, 1} are (+0, +1), (+1, +0), (−0, −1), (−1, −0).

The goal of this paper is to understand the Ehrhart theory of these four families of polytopes. Our main results are the following. Theorem 4.3 generalizes Stanley’s Theorem 1.1, offering combinatorial formulas for the Ehrhart quasipolynomials of the Coxeter permutahedra Π(A _n−1 ), Π(B _n ), Π(C _n ), and Π(D _n ) in terms of the combinatorics of forests. Theorems 5.2 and 5.3 then give explicit formulas: they compute the exponential generating functions of those Ehrhart quasipolynomials, in terms of the Lambert W function. Proposition 3.1 is an intermediate step that may be of independent interest: it describes the Ehrhart theory of a rational translate of an integral zonotope. This result was used in Ardila, et al. (To appear.) to compute the equivariant Ehrhart theory of the permutahedron.

We remark that each of these zonotopes can be translated to become an integral polytope, and the Ehrhart polynomials of these integral translates were computed in ^{Ardila, et al. (2015)}; see also ^{De Concini & Procesi (2008)}; ^{Deza, et al. (2018)} for related work.

2.1 Ehrhart theory

A rational polytope P ⊂ R2 is the convex hull of finitely many points in Q2. We define

for positive integers t. ^{Ehrhart (1962)} famously proved that this lattice-point counting function evaluates to a quasipolynomial in t, that is,

where c0t,…,cdt:Z⟶Q are periodic functions in t; their minimal common period is the period of ehr _P (t). Ehrhart also proved that the period of ehr _P (t) divides the least common multiple of the denominators of the vertex coordinates of P. In particular, if P is an integral polytope, then ehr _P (t) is a polynomial.

All the polytopes we will consider in this paper are half integral. Therefore the periods of their Ehrhart quasipolynomials will be either 1 or 2. For more on Ehrhart quasipolynomials, see, e.g., ^{Beck & Robins (2015)}.

2.2 Zonotopes

A zonotope is the Minkowski sum Z (A) of a finite set A = [a ₁ , b ₁],..., [a _n , b _n ] of line segments in R2; that is,

Equivalently, zonotopes are precisely the projections of cubes. For a finite set of vectors U⊂R2 we define

^{Shephard(1974)} showed that the zonotope Z (A) may be decomposed as a disjoint union of translates of the half-open parallelepipeds

spanned by the linearly independent subsets I of {b _j - a _j : 1≤ j ≤ n}. This decomposition contains exactly one parallelepiped for each independent subset. Figure 3 displays such a zonotopal decomposition of a hexagon.

A useful feature of this decomposition is that lattice half-open parallelepipeds are arithmetically quite simple: contains exactly vol( ) lattice points, where vol( ) denotes the relative volume of , measured with respect to the sublattice Zd∩ aff ( ) in the affine space spanned by the parallelepiped. This implies the following result.

Proposition 2.1. (^{Stanley (1991)}) Let U ⊂ Zdbe a finite set of vectors. Then the Ehrhart polynomial of the integral zonotope Z (U) is

whereWdenotes the number of vectors in W and vol(W) is the relative volume of the parallelepiped generated by W .

1.1 Lie combinatorics

Assuming familiarity with the combinatorics of Lie theory ^{Humphreys (1990)} (for this section only), we briefly explain the geometric origin of the polytopes that are our main objects of study. Finite root systems are highly symmetric configurations of vectors that play a central role in many areas of mathematics and physics, such as the classification of regular polytopes ^{Coxeter (1973)} and of semisimple Lie groups and Lie algebras Humphreys (1978). The finite crystallographic root systems can be completely classified; they come in four infinite families:

and five exceptions: E ₆ , E ₇ , E ₈ , F ₄ , and G ₂. For each of the four infinite families A _n−1 , B _n ,C _n , D _n of root systems Φ, we can let the positive roots Φ⁺ be those obtained by choosing the plus sign in each ± above.

Let Φ be a finite root system of rank d and W be its Weyl group. Let Φ⁺⊂ Φ be a choice of positive roots. The standard Coxeter permutahedron of Φ is the zonotope

where ρ := 12 (Σα∈Φ++ α). These polytopes, and their deformations, are fundamental objects in the representation theory of semisimple Lie algebras ^{Humphreys (1978)}, in many problems in optimization ^{Ardila, et al. (2020)}, and in the combinatorics of (signed) permutations, among other areas.

For the classical root systems A _n−1 , B _n ,C _n , D _n , the standard Coxeter permutahedra are precisely the polytopes Π(A _n−1 ), Π(B _n ), Π(C _n ), Π(D _n ) introduced in Section 1.2.

4.1 Signed graphs as a model for classical root systems

A signed graph G = (Γ, σ ) consists of a graph Γ = (V, E) and a signature σ ∈ {±}^E . The underlying graph Γ may have multiple edges, loops, halfedges (with only one endpoint), and loose edges (with no endpoints); the latter two have no signs. For the applications we have in mind, we may assume that G has no loose edges and no repeated signed edges; we do allow G to have two parallel edges with opposite signs.

A signed graph G = (Γ, σ) is balanced if each cycle has an even number of negative edges. An unsigned graph can be realized by a signed graph all of whose edges are labelled with +; it is automatically balanced.

Continuing a well-established dictionary ^{Zaslavsky (1981)}, we encode a subset S ⊆ Φ⁺ of positive roots of one of the classical root systems Φ ∈ {A _n−1 , B _n ,C _n , D _n : n ≥ 1} in the signed graph G _S on n nodes with

• a positive edge ij for each e _i − e _j ∈ S, • a halfedge at j for each e _j ∈ S, and

• a negative edge ij for each e _i + e _j ∈ S, • a negative loop at j for each 2e _j ∈ S.

The Φ-graphs are the signed graphs encoding the subsets of Φ⁺. More explicitly, a signed graph is an A _n−1 -graph (or simply a graph) if it contains only positive edges, a B _n -graph if it contains no loops, a C _n -graph if it contains no halfedges, and a D _n -graph if it contains neither halfedges nor loops. For a Φ-graph G, we let Φ _G ⊆ Φ⁺ be the corresponding set of positive roots of Φ.

It will be important to understand which subsets of Φ⁺ are linearly independent; to this end we make the following definitions.

A (signed) tree is a connected (signed) graph with no cycles, loops, or halfedges.

A (signed) halfedge-tree is a connected (signed) graph with no cycles or loops, and a single halfedge.

A (signed) loop-tree is a connected (signed) graph with no cycles or halfedges, and a single loop.

A (signed) pseudotree is a connected (signed) graph with no loops or halfedges that contains a single cycle (which is unbalanced).

A signed pseudoforest is a signed graph whose connected components are signed trees, signed halfedge-trees, signed loop-trees, or signed pseudotrees.

https://es.overleaf.com/project/5fd18252974026462556094e

A Φ-forest is a signed pseudoforest that is also Φ-graph for each of the root systems

Φ ∈ {A _n−1 , B _n ,C _n , D _n : n ≥ 1}.

A Φ-tree is a connected Φ-forest for Φ ∈ {A _n−1 , B _n ,C _n , D _n : n ≥ 1}.

In particular the A _n−1 -pseudoforests are the forests on [n] := {1, 2,..., n}. For a signed pseudoforest G, we let tc(G), hc(G), lc(G), and pc(G) be the number of tree components, halfedge-tree components, loop-tree components, and pseudotree components, respectively.

In this language, we recall and expand on results by ^{Zaslavsky (1982)} and ^{Ardila, et al. (2015)} on the arithmetic matroids of the classical root systems. Recall that for a linearly independent set W ⊂ Zd, we write vol(W) for the relative volume of the parallelepiped z (W) generated by W.

Proposition 4.1. ^{Ardila, et al. (2015 )}; Zaslavsky (1982) Let Φ ∈ {A _n-1 , B _n ,C _n , D _n } be a root system. The independent subsets of Φ⁺ are the sets Φ _G for the Φ-forests G on [n]. For each such G,

4.2 Ehrhart quasipolynomials of standard Coxeter permutahedron of classical type

We also define the integral Coxeter permutahedron

This is a translate of the standard Coxeter permutahedron Π(Φ) which is an integral polytope for all Φ. Its Ehrhart theory was computed in ^{Ardila, et al. (2015)}. This is sometimes, but not always, the same as the Ehrhart theory of Π(Φ), as we will see in this section, particularly in Theorem 4.3.

It follows from the description in Section 1.2 that the standard Coxeter permutahedron Π(Φ) is an integral polytope precisely for Φ ∈ {A _n−1 : n ≥ 1 odd} ∪ {C _n : n ≥ 1} ∪ {D _n : n ≥ 1}. It is shifted 121 := 12 (e1 + ··· + en) away from being integral for Φ ∈ {A _n−1 : n ≥ 2 even} ∪ {B _n : n ≥ 1}.

Proposition 4.2. Let Φ ∈ {A _n-1 : n ≥ 2 even} ∪ {B _n : n ≥ 1}. For a Φ-forest G, the affine subspace121 + span(Φ _G ) contains lattice points if and only if every (signed or unsigned) tree component of G has an even number of vertices.

Proof. Let G ₁ ,..., G _k be the connected components of G, on vertex sets V ₁ ,...,V _k, respectively. Along the decomposition Rn=Rv1⊕…⊕Rvk, we have

where 1 _V : = ∑ _i∈V e _i for V ⊆ [n]. Therefore 121+spanΦG contains a lattice point in Zn if and only if 121Vi+spanΦGi contains a lattice point in ZVi for every 1≤ I ≤ k. For this reason, it suffices to prove the proposition for Φ-trees.

For every labeling λ ∈ RE(G) of the edges of G with scalars, we will write

We need to show that for a Φ-tree G, there exists λ ∈ RE(G) with v _G (λ) ∈ Zn if and only if G is not a (signed or unsigned) tree with an odd number of vertices. We proceed by cases.

Trees: Let G = ([n], E) be a tree. If

is a lattice point for some choice of scalars λ = (λ _ij ) _ij∈E , then the sum of the coordinates of v _G (λ)-which ought to be an integer-equals 12n. Therefore n is even.

Conversely, suppose n is even. For each edge e = ij of G, let

We claim that v _G (λ), as defined in (4.2), is an integer vector. To see this, consider any vertex 1 ≤ m ≤ n and suppose that when we remove m and its adjacent edges, we are left with subtrees with vertex sets V ₁ ,...,V _k . Then

and this is an integer since ∑ki=1  |Vi| = n - 1 is odd.

We conclude that for a tree G, the affine subspace 121+spanΦG contains lattice points if and only if G has an even number of vertices, as desired.

Signed trees: Given a subset S ⊆ B _n = {±e _i ± e _j : 1 ≤ i < j ≤ n} ∪ {±e _i : 1≤ I ≤ n} , we define the vertex switching S _m of S at a vertex 1≤ m≤ n to be obtained by changing the sign of each occurrence of e _m in an element of S. Notice that the effect of this transformation on the expression

is simply to change the mth coordinate from 12 + a to 12 − a; this does not affect integrality.

Similarly, define the edge switching S _b of S at b ∈ S to be obtained by changing the sign of b in S. Notice that

where λ’ is obtained from λ by switching the sign of λ _s.

We conclude that vertex and edge switching a subset S ⊆ B _n does not affect whether 121+spanS intersects the lattice Zn. Now, it is known ^{Zaslavsky (1982)} that for any balanced signed graph G there is an ordinary graph H such that Φ _G can be obtained from Φ _H by vertex and edge switching. In particular-as can also be checked directly-any signed tree G can be turned into an unsigned tree H in this way. Invoking case (i) for the tree H, we conclude that for a signed tree G, 121+spanΦG contains lattice points if and only if G has an even number of vertices.

Signed halfedge-trees: Let G be a signed halfedge tree. We need to show that 121+spanΦG contains a lattice point. Let h be the halfedge. There are two cases:

a. If n is even, we can label the edges s of G ⁻:= G − h with scalars λ _s in such a way that v _G − (λ | _G- ) ∈ Zn, in view of (ii). Setting the weight of the halfedge λ _h = 0 we obtain v _G (λ | _G )= v _G − (λ | _G- ) ∈ Zn, as desired.

b. If n is odd, let G ⁺ be the signed tree obtained by turning the half edge h into a full edge h ⁺, going to a new vertex n + 1. Using (ii), we can label the edges s of G+ with scalars λ _s such that v _G+ (λ | _G+ ) ∈ Zn+1. Setting the weight of the halfedge h in G to be λ _h = λ _h+ , we obtain that v _G (λ | _G ) is obtained from v _G+ (λ | _G+ ) by dropping the last coordinate; therefore v _G (λ | _G ) ∈ Zn as desired.

Signed pseudotrees: Let G be a signed pseudotree. We need to find scalars λ _s such that v _G (λ ) is a lattice vector. Assume, without loss of generality, that its unique (unbalanced) cycle C is formed by the vertices 1,..., m in that order. Let T ₁ ,..., T _k be the subtrees of G hanging from cycle C; say T _i is rooted at the vertex a _i , where 1≤ a _i ≤ m, and let s _i be the edge of T _i connected to a _i . We find the scalars λ _s in three steps.

1. Thanks to (ii), for each tree T _i with an even number of vertices, we can label its edges s with scalars λ _s such that

2. For each tree T _i with an odd number of vertices, we can label the edges s of T _i − s _i with scalars λ _s such that vTi-siλTi-si=121Vi-ai+∑s∈ETi-siλss∈ZVi-ai. Setting λsi = 0, we obtain

3. It remains to choose the scalars λ ₁₂ ,..., λ _m1 corresponding to the edges of the cycle C. Since E(G) is the disjoint union of E(C) and the E(T _i )s, we have

is supported on the vertices [m]= {1,..., m} of the cycle C. Therefore, v _G (λ ) ∈ Zn if and only if we have v _C (λ | _C )+ t ∈ Zm, where t:u+12∑i:Vieveneai. We rewrite this condition as

where σ _i is the sign of edge connecting i and i + 1 in C; this is equivalent to the following system of equations modulo 1:

Solving for λ ₁₂ gives λ12≡σ1…σmλ12 + a for a scalar a. Since the cycle C is unbalanced, σ1…σm=-1, so this equation has the solution λ12≡ a/2 (mod 1). Using (4.4), we can then successively compute the values of λ ₂₃ ,..., λ _m1 , guaranteeing that (4.3) holds. In turn, this produces a lattice point vGλ∈ 121+spanΦG, as desired.

Theorem 4.3. Let Z (Φ) be the set of Φ-forests, and E (Φ) ⊆ F (Φ) be the set of Φ-forests such that every (signed) tree component has an even number of vertices.

1. The Ehrhart polynomials of the integral Coxeter permutahedraΠZ(Φ)are

For Φ ∈ {A _n-1 : n ≥ 2 even} ∪ { B _n : n ≥ 1} , the Ehrhart quasipolynomials of the standard Coxeter permutahedra Π(Φ) are

Proof. This is the result of applying Propositio 3.1 to these zonotopes, taking into account Propositions 4.1 and 4.2, and the fact that Φ-forests of type A and B contain no loop components.

5.1 Tree enumeration

Proposition 5.1. The enumeration of (signed) trees, (signed) pseudotrees, signed halfedgetrees, and signed loop-trees is given by the following formulas.

The number of trees on [n] is t _n = n ⁿ⁻² . The exponential generating function for this sequence is

The number of pseudotrees on [n] is p _n , where

The number of signed trees on [n] is st _n = 2 ⁿ⁻¹ n ⁿ⁻² . The exponential generating function for this sequence is

4. The number of signed pseudotrees on [n] is sp _n , where

The number of signed half-edge trees on [n] and of signed loop-trees is sh _n = sl _n = (2n) ⁿ⁻¹ . The exponential generating function for this sequence is

Proof. We begin by remarking that most of these formulas were obtained by Vladeta Jovovic and posted without proof in entries A000272, A057500, A097629, A320064, and A052746 of the Online Encyclopedia of Integer Sequences Sloane (n.d.). For completeness, we provide proofs.

The formula for t _n is well known and due to Cayley; see for example (^{Stanley, 1999}, Proposition 5.3.2). Now, by the multiplicative formula for exponential generating functions (Stanley, 1999, Proposition 5.1.1), W (-x)² /2 is the generating function for pairs of rooted trees (T ₁ , r ₁) and (T ₂ , r ₂), the disjoint union of whose vertex sets is [n]. By adding an edge between r ₁ and r ₂, we see that this is equivalent to having a single tree with a special chosen edge r ₁ r ₂; there are n ⁿ⁻² (n − 1) such objects. Therefore

proving the desired generating function.

A pseudotree on [n] is equivalent to a choice of rooted trees (T ₁ , r ₁),..., (T _k , r _k ), the union of whose vertex sets is [n], together with a choice of an undirected cyclic order on r ₁ ,..., r _n - or equivalently, an undirected cyclic order on those trees. Since the exponential function for rooted trees and for undirected cyclic orders are −W (−x) and

respectively, the desired result follows by the compositional formula for exponential generating functions.

There are 2 ⁿ⁻¹ choices of signs for a tree on [n], so we have st _n = 2 ⁿ⁻¹ t _n . Combining with 1. gives the desired formulas.

Each pseudotree on [n] can be given 2 ⁿ different edge sign patterns, half of which will lead to an unbalanced cycle; this leads to 2 ⁿ⁻¹ p _n signed pseudotrees. This accounts for all signed pseudotrees, except for the ones containing a 2-cycle. We obtain such an object by starting with a signed tree, choosing one of its edges, and inserting the same edge with the opposite sign. This counts each such object twice, so the total number of them is st _n (n-1)/2. It follows that sp _n = 2 ⁿ⁻¹ p _n + st _n (n-1)/2, from which the desired formulas follow using 2. and 3.

A signed half-edge tree (or a signed loop-tree) is obtained from a signed tree by choosing the vertex where we will attach the half-edge (or loop). Thus sh _n = sl _n = n · st _n = (2n) ⁿ⁻¹ . The exponential generating function follows directly from the definition of W (x).

5.2 Generating functions of Ehrhart (quasi)polynomials of Coxeter permutahedra

Theorem 5.2. The generating functions for the Ehrhart polynomials of the integral Coxeter permutahedra of the classical root systems are:

Proof. Theorem 4.3.1 tells us that these exponential generating functions can be understood as enumerating various families of (pseudo)forests, weighted by their various types of connected components. The compositional formula for exponential generating functions (^{Stanley, 1999}, Theorem 5.1.4) then expresses them in terms of the exponential generating functions for each type of connected component.

For example, in type A there are only tree components, so

by Proposition 5.1.1.

Similarly, for the other types we have

and, analogously,

Carefully substituting the formulas in Proposition 5.1, we obtain the desired results.

Using the formulas in Theorem 5.2 and suitable mathematical software, one easily computes the following table of Ehrhart polynomials. The reader may find it instructive to compare this with the analogous table in (^{Ardila, et al. 2015}, Section 6), which lists the Ehrhart polynomials with respect to the weight lattice of each root system. The tables coincide only in type C, which is the only classical type where the weight lattice is Zn

Theorem 5.3. The generating function for the odd part of the Ehrhart quasipolynomials of the non-integral standard Coxeter permutahedra are the following. For t odd,

Proof. We carry out similar computations as for Theorem 5.2. This requires us to observe that the generating functions for even trees and even signed trees are

Now, in light of Theorem 4.3.2, and analogously to the proof of Theorem 5.2, we have

and

which give the desired results using Proposition 5.1.

Using these formulas, and combining them with Table 1, one computes the following table of Ehrhart quasipolynomials.

The reader may find it instructive to count the lattice points in the polygons of figure 1, and compare those numbers with the predictions given by tables 1 and 2.