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Revista de la Academia Colombiana de Ciencias Exactas, Físicas y Naturales

Print version ISSN 0370-3908

Rev. acad. colomb. cienc. exact. fis. nat. vol.44 no.173 Bogotá Oct./Dec. 2020  Epub July 08, 2021

https://doi.org/10.18257/raccefyn.1189 

Mathematics

The arithmetic of Coxeter permutahedra

La aritmética de los permutaedros de Coxeter

Federico Ardila1  2  * 

Matthias Beck1  3 

Jodi McWhirter1  4 

1 San Francisco State University, San Francisco, California, USA.

2 Universidad de Los Andes, Bogotá, Colombia.

3 Freie Universität, Berlin, Germany.

4 Washington University in St. Louis, Missouri, USA.


Abstract

Ehrhart theory measures a polytope P discretely by counting the lattice points inside its dilates P, 2P, 3P, We compute the Ehrhart theory of four families of polytopes of great importance in several areas of mathematics: the standard Coxeter permutahedra for the classical Coxeter groups A n , B n , C n , Dn. A central tool, of independent interest, is a description of the Ehrhart theory of a rational translate of an integer projection of a cube.

Keywords: Polytope; Ehrhart theory; Coxeter group; Permutahedron; Tree; Lambert function

Resumen

La teoría de Ehrhart mide un politopo P discretamente, contando los puntos enteros dentro de sus dilataciones P, 2P, 3P, En este artículo calculamos la teoría de Ehrhart de cuatro familias de politopos de gran importancia en varias áreas de la matemática: los permutaedros de Coxeter de los grupos clásicos de Coxeter A n , B n , C n , D n . Una herramienta central, de interés independiente, es la descripción de la teoría de Ehrhart de una traslación racional de una proyección entera de un cubo.

Palabras clave: Politopo; Teoría de Ehrhart; Grupo de Coxeter; Permutaedro; Árbol; Función de Lambert

1. Introduction

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.

Figure 1 The first three dilates of the standard Coxeter permutahedra Π(A 2), Π(B 2), Π(C 2), and Π(D 2). Their tth dilates contain 1 + 3t + 3t 2, (1 + 4t + 7t 2 for t even and 2t + 7t 2 for t odd), 1 + 6t + 14t 2, and 1 + 2t + 2t 2 lattice points, respectively. 

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.

Figure 2 The permutahedron Π4 organizes the 24 permutations of {1, 2, 3, 4}. 

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

ehrΠnt=an-1-tn-1+an-2tn-2++a1t+a0

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:

Π(An-1)  :=  conv{permutations of 12 (-n + 1, -n + 3,..., n - 3, n - 1)},

Π(Bn)  :=  conv{permutations of 12 (1, 3,..., 2n - 1)},

Π(Cn) := conv{signed permutations of (1, 2,..., n)},

Π(Dn) := conv{evenly signed permutations of (0, 1,..., n - 1)}.

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. Preliminaries

2.1 Ehrhart theory

A rational polytope PR2 is the convex hull of finitely many points in Q2. We define

ehrpttPZd,

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

ehrpt=cdttd+cd-1ttd-1+c0t

where c0t,,cdt:ZQ 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 UR2 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.

Figure 3 A decomposition of a hexagon into half-open parallelepipeds. 

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:

An-1±ei-ej:1i<jn,

Bn±ei-ej,±ei+ej:1i<jn±ei:1in,

Cn±ei-ej,±ei+ej:1i<jn±2ei:1in,

Dn±ei-ej,±ei+ej:1i<jn

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

ΠΦ      :=     αΦ+-α2,α2

=      conv{wρ:wW}

where ρ := 12 (ΣαΦ++ 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.

3 Almost integral zonotopes and their Ehrhart theory

The arithmetic of zonotopes described in Section 2.2 becomes much more subtle when the zonotope is not integral. However, we can still describe it for almost integral zonotopes v + Z (U v. They satisfy the following analog of Stanley’s Propositi), which are obtained by translating an integral zonotope Z (U) by a rational vector on 2.1.

Proposition 3.1. LetUZdbe a finite set of integer vectors andvQdbe a rational vector. Then the Ehrhart quasipolynomial of the almost integral zonotope v +Z (U) equals

where

Xwt  :=1 if (tv+span(W))Zd,0         otherwise                            

Proof. The zonotope t(v + Z (U)) can be subdivided into lattice translates of the half-open parallelepipeds for the linearly independent subsets WU. Let us count the lattice points in ; there are two cases:

1. If t v + span(W) does not intersect Zd then

2. If t v + span(W) contains a lattice point UZd, then it also contains the lattice points u + w for all wW, so Λ := (t v + span(W)) ∩ Zd is a |W|-dimensional lattice. Since t v + span(W) can be tiled by integer translates of the half-open parallelepiped , and that linear space contains the lattice Λ, each tile must contain lattice points. Therefore

and the desired result follows.

In Ardila, et al. (To appear.), Proposition 3.1 is used to describe the equivariant Ehrhart theory of the permutahedron and prove a series of conjectures due to Stapledon (2011) in this special case.

4 Classical root systems, signed graphs and Ehrhart functions

We will express the Ehrhart quasipolynomials of the classical Coxeter permutahedra in terms of the combinatorics of signed graphs. These objects originated in the social sciences and have found applications also in biology, physics, computer science, and economics; they are a very useful combinatorial model for the classical root systems. See Zaslavsky (1998) for a comprehensive bibliography.

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 WZd, 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,

ΦG=n-tcG                and           volΦG=2pcG+lc(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=Rv1Rvk, we have

121+spanΦG=i=1k121Vi+spanΦGi

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 λRE(G) of the edges of G with scalars, we will write

vGλ121+sE(G)λss.  (4.1)

We need to show that for a Φ-tree G, there exists λRE(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

vGλ121+ijE(G)λijei-ej.  (4.2)

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

λij =0  if G-e consists of two subgraphs with an even number of vertices each, and12 G-e consists of two subgraphs with an odd number of vertices each.              

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

vG(λ)m  12 + 12 (number of 1  i  k such that |Vi| is odd)     (mod 1),

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

121+sSλss

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

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

121+sSλss=121+sSbλs's

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

We conclude that vertex and edge switching a subset SB 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 := Gh 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

vTiλTiZVi.

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+sETi-siλssZVi-ai. Setting λsi = 0, we obtain

vTiλTi12eai+ZVi.

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

vGλ=vCλc+i=1kvTiλTi+u,              where          u=121-1m-i=1k1Vi Rm

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 )+ tZm, where t:u+12i:Vieveneai. We rewrite this condition as

λ12e1 - σ1e2+ λ23e2 - σ2e3+ ··· + λm1em - σme1+ t  Zm, (4.3)

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:

λ12  λm1σm -t1,   λ23  λ12σ1 -t2,   . ..,λm1  λm-1,mσm-1-tm          (mod 1). (4.4)

Solving for λ 12 gives λ12σ1σmλ12 + a for a scalar a. Since the cycle C is unbalanced, σ1σm=-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 Explicit formulas: the generating functions

In this section, we compute the generating functions for the Ehrhart (quasi)polynomials of the Coxeter permutahedra of the classical root systems. We will express them in terms of the Lambert W function

Wx=n1(-n)n-1xnn!.

As a function of a complex variable x, this is the principal branch of the inverse function of xe x . It satisfies

Wx=eW(x)=x.

Combinatorially, -W (-x) is the exponential generating function for r n = n n−1 , the number of rooted trees (T, r) on [n], where T is a tree on [n] and r is a special vertex called the root (Stanley, 1999, Proposition 5.3.2).

To compute the generating functions of the Ehrhart (quasi)polynomials that interest us, we first need some enumerative results on trees.

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

Tx:=n1nn-2xnn!=-W-x-12W-x2

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

Px:=n1pn-2xnn!=12W-x-14W-x2-12log1+W-x.

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

STx:=n12n-1nn-2xnn!=-12W-2x-14W-2x2.

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

SPx:=n1spnxnn!=-14W-2x-log1+W(-2x).

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

SHx=SLxn1(2)n-1xnn!=-12W-2x.

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

12W-x2=n0nn-2n-1xnn!=-W-x-Tx,

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

x-x22+n3n-1!2xnn!=x2+x24+12log1-x,

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:

1+n1ehrΠZAn-1txnn!=exp-1tW-tx-12tW-tx2,

1+n1ehrΠZBntxnn!=exp-12tW-2tx-14tW-2tx2 /1+W-2tx,

1+n1ehrΠZCntxnn!=exp-t-12tW-2tx-14tW-2tx2/ 1+W-2tx,

1+x+n2ehrΠZDntxnn!=expt-12tW-2tx-14tW(-2tx)2/ 1+W-2tx.

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

1+n1ehrΠZAn-1txnn!=n0forestG on n tn-tc(G)xnn!

=n0forestG on n 1ttcGtxnn!

=exp1tn0treesT on n txnn!

=exp1tT(tx)

=exp-1tW-tx-12tW(-tx)2

by Proposition 5.1.1.

Similarly, for the other types we have

1+n1ehrΠZBntxnn!=n0B-forestG on n 2pcGtn-tcGxnn!

=n0B-forestG on n 2pcG1ttcG1hcGtxnn!

=exp2SPtx+1tSTtx+SH(tx)

and, analogously,

1+n1ehrΠZCntxnn!=exp2SPtx+1tSTtx+2SLtx,

1+x+n2ehrΠZDntxnn!=exp2SPtx+1tSTtx,

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

Table 1 Ehrhart polynomials of integral Coxeter permutahedra. 

Φ Ehrhart polynomial of ΠZ(Φ+)
A 0 1
A 1 1 + t
A 2 1 + 3t + 3t 2
A 3 1 + 6t + 15t 2 + 16t 3
B 1 1 + t
B 2 1 + 4t + 7t 2
B 3 1 + 9t + 39t 2 + 87t 3
B 4 1 + 16t + 126t 2 + 608t 3 + 1553t 4
C 1 1 + 2t
C 2 1 + 6t + 14t 2
C 3 1 + 12t + 66t 2 + 172t 3
C 4 1 + 20t + 192t 2 + 1080t 3 + 3036t 4
D 2 1 + 2t + 2t 2
D 3 1 + 6t + 18t 2 + 32t 3
D 4 1 + 12t + 72t 2 + 280t 3 + 636t 4

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,

1+n1ehrΠA2n-1tx2n(2n)!=exp-W-tx+W(tx)2t-W-tx2+W(tx)24t

1+n1ehrΠBntxnn!=exp-W-2tx+W(2tx)4t-W-2tx2+W(2tx)28t1+W(-2tx)

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

Tevenxn0t2nx2nn!=12T(x)+T-x,

STevenxn0st2nx2nn!=12ST(x)+ST-x.

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

1+n1ehrΠA2n-1tx2n2n!=exp1tTeventx

=exp12tTtx+12tT-tx

and

1+n1ehrΠBntxnn!=exp2SPtx+1tSTeventx+2SLtx

=exp 2SPtx+12tSTtx+12tST-tx+2SLtx,

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.

Table 2 Ehrhart quasipolynomials of the non-integral standard Coxeter permutahedra. 

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.

6. Acknowledgments

Some of the results of this paper are part of the Master’s theses of JM at San Francisco State University, under the supervision of FA and MB McWhirter (2019). We would like to thank Mariel Supina and Andre´s Vindas-Meléndez for valuable discussions, and Jean-Philippe Labbé for checking our computations of the Ehrhart (quasi)polynomials of tables 1 and 2. This paper was written while FA was on sabbatical at the Universidad de Los Andes in Bogotá. He thanks Los Andes for their hospitality and SFSU and the Simons Foundation for their financial support. He was supported by National Science Foundation grant DMS-1855610 and Simons Fellowship 613384.

References

Ardila, F., Castillo, F., Eur, C., & Postnikov, A. (2020). Coxeter submodular functions and deformations of Coxeter permutahedra. Adv. Math. 365: 107039. [ Links ]

Ardila, F., Castillo, F., & Henley, M. (2015). The arithmetic Tutte polynomials of the classical root systems. International Mathematics Research Notices. 2015 (12): 3830-3877. [ Links ]

Ardila, F., Supina, M., & Vindas-Meléndez, A. R. (To appear). The equivariant Ehrhart theory of the permutahedron. Proceedings of the American Mathematical Society. (arXiv:1911.11159) [ Links ]

Bárány, I., & Füredi, Z. (1987). Computing the volume is difficult. Discrete & Computational Geometry. 2 (4): 319-326. [ Links ]

Beck, M., & Robins, S. (2015). Computing the Continuous Discretely: Integer-point Enumeration in Polyhedra (Second ed.). Springer, New York. Retrieved from http://dx.doi.org/10.1007/978-1-4939-2969-6 (electronically available at http://math.sfsu.edu/beck/ccd.html) doi: 10.1007/978-1-4939-2969-6 [ Links ]

Coxeter, H. S. M. (1973). Regular Polytopes. Courier Corporation. [ Links ]

De Concini, C., & Procesi, C. (2008). The zonotope of a root system. Transform. Groups. 13 (3-4): 507-526. [ Links ]

Deza, A., Manoussakis, G., & Onn, S. (2018). Primitive zonotopes. Discrete Comput. Geom. 60 (1): 27-39. [ Links ]

Dyer, M. E., & Frieze, A. M. (1988). On the complexity of computing the volume of a polyhedron. SIAM Journal on Computing. 17 (5): 967-974. [ Links ]

Ehrhart, E. (1962). Sur les polyèdres rationnels homothétiques à n dimensions. C. R. Acad. Sci. Paris. 254: 616-618. [ Links ]

Fukuda, K. (2008). Software package cdd. (Electronically available at http://www.ifor.math.ethz.ch/fukuda/) [ Links ]

Humphreys, J. E. (1978). Introduction to Lie Algebras and Representation Theory (Vol. 9). Springer-Verlag, New York-Berlin. (Second printing, revised) [ Links ]

Humphreys, J. E. (1990). Reflection Groups and Coxeter Groups (Vol. 29). Cambridge University Press, Cambridge. Retrieved from https://doi.org/10.1017/CBO9780511623646 doi: 10.1017/CBO9780511623646 [ Links ]

McWhirter, J. (2019). Ehrhart quasipolynomials of Coxeter permutahedra (Master’s thesis, San Francisco State University). https://sfsu-dspace.calstate.edu/bitstream/handle/10211.3/213961/AS362019MATHM39.pdf?sequence=1. [ Links ]

Shephard, G. C. (1974). Combinatorial properties of associated zonotopes. Canad. J. Math. 26 (2): 302-321. [ Links ]

Sloane, N. (n.d.). The On-Line Encyclopedia of Integer Sequences. (published electronically at http://oeis.org, 2014) [ Links ]

Stanley, R. P. (1991). A zonotope associated with graphical degree sequences. In Applied geometry and discrete mathematics (Vol. 4, pp. 555-570). Providence, RI: Amer. Math. Soc. [ Links ]

Stanley, R. P. (1999). Enumerative Combinatorics . Volume 2 (Vol. 62). Cambridge: Cambridge University Press. (With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin) [ Links ]

Stapledon, A. (2011). Equivariant Ehrhart theory. Advances in Mathematics. 226 (4): 3622-3654. [ Links ]

Zaslavsky, T. (1981). The geometry of root systems and signed graphs. Amer. Math. Monthly. 88 (2): 88-105. [ Links ]

Zaslavsky, T. (1982). Signed graphs. Discrete Appl. Math. 4 (1): 47-74. [ Links ]

Zaslavsky, T. (1998). A mathematical bibliography of signed and gain graphs and allied areas. Electron. J. Combin. 5. Dynamic Surveys 8, 124 pp. (Electronically available at http://www.math.binghamton.edu/zaslav/Bsg/index.html) [ Links ]

Federico Ardila, Inaugural article for admission as a Corresponding Member to the Academia Colombiana de Ciencias Exactas, Físicas y Naturales

Citation: Ardila F, Beck M, McWhirter J. The arithmetic of Coxeter permutahedra. Rev. Acad. Colomb. Cienc. Ex. Fis. Nat. 44(173):1152-1166, octubre-diciembre de 2020. doi: https://doi.org/10.18257/raccefyn.1189

Editor: Jorge Cossio

1 In fact it has exactly two solutions λ 12a/2 (mod 1) and λ 12 ≡ (1 + a)/2 (mod 1), explaining why we have vol(Φ G )= 2 in this case.

Received: April 04, 2020; Accepted: July 14, 2020; Published: December 05, 2020

*Corresponding author: Federico Ardila; federico@sfsu.edu

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