Compositions constrained by graph Laplacian minors

Motivated by examples of symmetrically constrained compositions, super convex partitions, and super convex compositions, we initiate the study of partitions and compositions constrained by graph Laplacian minors. We provide a complete description of the multivariate generating functions for such compositions in the case of trees. We answer a question due to Corteel, Savage, and Wilf regarding super convex compositions, which we describe as compositions constrained by Laplacian minors for cycles; we extend this solution to the study of compositions constrained by Laplacian minors of leafed cycles. Connections are established and conjectured between compositions constrained by Laplacian minors of leafed cycles of prime length and algebraic/combinatorial properties of reflexive simplices.


Introduction
A partition of a positive integer n is a sequence λ := (λ 1 , λ 2 , . . . , λ k ) of non-negative integers satisfying λ k ≤ λ k−1 ≤ · · · ≤ λ 1 such that i λ i = n. We call each λ i a part of λ and say that λ has k parts. A (weak) composition of n is a sequence λ := (λ 1 , λ 2 , . . . , λ k ) of nonnegative integers satisfying i λ i = n. Beginning with [1] and through eleven subsequent papers, George Andrews and various coauthors successfully revived Percy MacMahon's technique of Partition Analysis (PA) from obscurity as a tool for computing generating functions for partitions and compositions. PA is particularly well-suited to the study of partitions and compositions constrained by a linear system of inequalities, i.e., of the form Aλ ≥ b for some integral matrix A and integral vector b; this is equivalent to the study of integer points in a rational polyhedral cone. Motivated in part by this renewed interest in PA, there has been The connection between the incidence matrix of a graph and its Laplacian is the wellknown equation L = ∂∂ T [8, Chapter 6]. Kirchoff's Matrix-Tree Theorem [8] asserts that given a graph G on {0, . . . , n} and given any i between 0 and n, det(L i ) is equal to the number of labeled spanning trees of G. This establishes a strong connection between Laplacian minors and enumerative properties of graphs.  in Example 3 of [12]; they were first studied using techniques from MacMahon's Partition Analysis by Andrews in [1], where they are named partitions with mixed difference conditions.
The multivariate generating function for such partitions is described in [12].  [12].
We now introduce our primary object of study. Definition 1.6. Given a graph G with Laplacian L and a vertex i of G, the compositions constrained by the i-th Laplacian minor of G are those satisfying L i λ ≥ 0.
Our paper is structured as follows. In Section 2, we review techniques from integer point enumeration in polyhedral cones that will be useful for producing generating function identities. In Section 3, we investigate compositions constrained by Laplacian minors for trees. Theorem 3.1 gives a combinatorial interpretation for the entries of the matrix inverse for tree Laplacian minors, leading to Corollary 3.2, the main result of the section.
In Section 4, we provide a solution to the open problem of Corteel, Savage, and Wilf discussed in Example 1.5. However, as shown in the section, our solution is less clear than 3 one might want. Thus, motivated by the underlying graphical structure of our compositions, in Section 5 we examine compositions constrained by Laplacian minors for leafed cycles, i.e.
for cycles with an additional vertex forming a leaf appended. These compositions are closely related to the super convex compositions in Example 1.5. The addition of a leaf to the cycle causes a change in the structure of the resulting compositions, leading to Theorem 5.1. We also consider in Conjecture 5.5 a possible combinatorial interpretation for the compositions constrained by leafed cycles, and we prove special cases of this conjecture.
In Section 6, we consider generating functions for certain subfamilies of compositions constrained by Laplacian minors for leafed cycles of prime length. Surprisingly, these generating functions are Ehrhart series for a family of reflexive simplices, as shown in Theorem 6.3. After introducing Ehrhart series and reflexive polytopes and proving Theorem 6.3, we indicate some possible connections between our enumerative problem and the algebraic/combinatorial structure of these simplices.
We close in Section 7 with a conjecture regarding compositions constrained by Laplacian minors for leafed cycles of length 2 k .

Cones and generating functions
The problem of enumerating integer compositions and partitions satisfying a system of linear constraints Aλ ≥ b is identical to that of enumerating integer points in the polytope or polyhedral cone defined by these constraints. For further background related to integer point enumeration in polyhedra, see [6].
Given a square matrix A with det(A) = 0, the solution set is called an n-dimensional simplicial cone. The reason for the terminology "simplicial" comes from C being an n-dimensional cone with n bounding hyperplanes, also called facets.
The following lemma, whose proof is straightforward, indicates that simplicial cones can be described either by specifying these n facets or by specifying the n ray generators for the cone.
Lemma 2.1. Let A be an invertible matrix in R n×n . Then where each inequality is understood componentwise. 4 We encode the integer points in a cone C using the integer point transform of C given by the multivariate generating function This function encodes the integer points in C as a formal sum of monomials. Throughout this paper, we will typically need to specialize σ C in one of two ways. If we set z 1 = z 2 = · · · = z n = q, we obtain where |m| = m 1 + m 2 + · · · + m n . Setting z 1 = q and z 2 = z 3 = · · · = z n = 1, we obtain When A is an integer matrix having det(A) = 1, the ray generating matrix A −1 for the ndimensional simplicial cone C = {x ∈ R n : Ax ≥ 0} has integer entries; hence, every integer point in C can be expressed as A −1 y for a unique integer vector y ≥ 0. In this case, the integer point transform can be simplified as a product of geometric series as follows.
Lemma 2.2. Let A = (a i,j ) be an integer matrix with det(A) = 1 and inverse matrix More generally, suppose det(A) = 1. In this case, σ C (z 1 , z 2 , . . . , z n ) is still rational, but A −1 need not have integer entries. As a result, representations of integer points in C are not necessarily obtained by using integral combinations of the minimal integral vectors on the generating rays of C. We can work around this difficulty in the following manner.
Let v 1 , v 2 , . . . , v n be the (integral) ray generators of C. Define the (half-open) fundamental parallelepiped (FPP) to be The cone C can be tiled by integer translates of Π, but there may be integer points contained in Π that can only be obtained using strictly rational scaling factors of the ray generators of C. This leads to our main tool for computing generating functions. 5 Lemma 2.3. Using the preceding notation, if the points contained in Π are w 1 , w 2 , . . . , w m , the integer point transform of C is given by

Trees
We begin our study of compositions constrained by graph Laplacian minors with the special case of trees. Since a tree T has a unique spanning tree, namely itself, the Matrix-Tree Theorem implies that det(L i ) = 1 for any vertex i. Thus, compositions constrained by tree Laplacian minors will have a generating function of the form .
Since the b k,j 's arise as the entries in the inverse matrix for L i , we are interested in calculating this inverse. We first observe that without loss of generality we may consider only those vertices i which are leaves of T , through the following reduction.
Suppose that a vertex i ∈ T is not a leaf. Then taking the minor at i deletes row and column i from the Laplacian matrix, which equates to removing vertex i in the graph. We note that removing the vertex i will split our tree into some number of disjoint subtrees, which we label s 1 , . . . , s k . Label the vertices of s 1 consecutively from r 1 + 1, r 1 + 2, . . . , r 2 , then label the vertices of s 2 consecutively as r 2 + 1, r 2 + 2 . . . , r 3 , and so on until labeling the vertices of s k as r k + 1, . . . , r k+1 . Now consider the r k+1 × r k+1 matrix M, indexed naturally.
M has a block form resulting from the fact that the subtrees s 1 , . . . , s k are distinct and disconnected. In particular, each subtree s j , 1 ≤ j ≤ k, will have a corresponding block matrix M j consisting of rows r j + 1, . . . , r j+1 and columns r j + 1, . . . , r j+1 .
Further, each M j will correspond to the Laplacian minor for s j where the minor is taken at i, which is now simply a leaf of s j . By properties of block matrices, we have that M −1 will be a block matrix with blocks M −1 j . Thus, to find M −1 , we compute the Laplacian minor inverse for each subtree. With this observation, we can begin our search for a general form of the generating function of compositions constrained by Laplacian minors of trees.

Inverses of Laplacian minors for trees.
To determine the denominator of σ C for an arbitrary tree T minored at a leaf, we first find a combinatorial interpretation for the columns of the inverse of the Laplacian minor for T . (While Theorem 3.1 below seems likely to be known to experts in algebraic graph theory, we could not find an explicit statement of it in the literature.) Throughout this section, we assume that T is a tree on the vertex set {0, . . . , n} with an arbitrary orientation of the edges of T .
Theorem 3.1. Consider a tree T and its corresponding Laplacian matrix L. Minor at a leaf n of T and let L −1 n = (l i,j ). Then entry l i,j is the distance from n to the path connecting the vertices i and j. Equivalently, l i,j is the length of the intersection of the path from n to i with the path from n to j.
Theorem 3.1 has the following implication on the level of generating functions. Let σ C (q) denote the specialiazed generating function σ C (q, q, . . . , q).
Corollary 3.2. The specialized generating function for a tree T minored at leaf n is given by where b i is the sum of the distances between n and the paths connecting vertex i with the other vertices of T .
To prove Theorem 3.1, we return to the fact that L = ∂∂ T and proceed combinatorially.
We say the n-th subminor of ∂ (resp. of ∂ T ) imposed by a vertex n of a tree is the square matrix in which the n-th row, (resp. column) is deleted. We denote the n-th subminor of where g e,j is determined as follows: • 1 if edge e is on the path between j and n and e points away from n; • −1 if edge e is on the path between j and n and e points toward n; • 0 if edge e is not on the path between j and n.
Then G n ∂ n = I, hence ∂ −1 n = G n .
n , we may multiply columns i and j in ∂ −1 n component-wise to find l i,j . Consider an arbitrary edge of the graph. If the edge belongs to only one of the paths from i to n or from j to n, then there will be a zero corresponding to that edge in either column i or j, contributing nothing to entry l i,j . Otherwise we obtain a 1, since the sign of a row for any edge is constant. Thus, summing the component-wise product will record the length of the path from n to the path between i and j.
Theorem 3.1 provides some insight into the form of the generating function for compositions constrained by an arbitrary tree. However, determining the lengths of paths between vertices in trees is not an easy exercise in general. In the next subsection, we analyze a special case.
3.2. k-ary trees. We next provide an explicit formula for a specialized generating function for compositions constrained by Laplacian minors of k-ary trees. Recall that a k-ary tree is a tree formed from a root node by adding k leaves to the root, then adding k leaves to each of these vertices, etc., stopping after a finite number of iterations of this process. To study the compositions constrained by a Laplacian minor of a k-ary tree T , we add a leaf labeled 0 to the root of T and minor at this new vertex. We define the level of a vertex as its distance to vertex 0 and a subtree of a vertex as the tree which includes the vertex and all of its children (we assume all edges are oriented away from vertex 0). Recall that [n] q := 1 + q + q 2 + q 3 + · · · + q n−1 .
8 Theorem 3.4. For a k-ary tree with n levels, Proof. Suppose a k-ary tree T is given. In order to find the generating function consider the column sums of L −1 0 . Choose a particular column which will correspond to a vertex v ∈ T . The column sum will be the sum of the distances between 0 and all distinct paths in the tree with v as an endpoint. By the symmetry of k-ary trees, we choose an arbitrary vertex v at level j ∈ [n], since the corresponding column sum will be the same as any other vertex on that level.
is the column sum corresponding to any vertex at level j. For any vertex c which is a child of v (and including v), the distance between 0 and the path from v to c will be j. There are such vertices and each of these vertices contributes j to the column sum. Hence in total these vertices contribute the first term to our claimed exponent.
Consider the path between 0 and v, define S i to be the set of vertices which are in the subtree with w i as the root but not in the subtree with w i−1 as the root. The vertex w i is on level j − i of T . Thus, the subtree whose root is w i has n − (j − i) levels of its own. So, there are k 0 + k 1 + · · · + k n−(j−i) vertices in the subtree with w i as a root, and k 0 + k 1 + · · · + k n−(j−i)−1 vertices in the subtree of w i−1 .
Thus, there are vertices in S i . By definition, the shortest distance between 0 and the path connecting v and any vertex s ∈ S i is j − i, since the path connecting v and s passes through w i , which must be the closest point on this path to 0.
Therefore, for each i ∈ [j − 1], the sum of the columns corresponding to vertices in S i will be (j − i)(k n−(j−i) ). So, for all vertices which are not children of v, the total column sums is the column sum 9 corresponding to any vertex at level j ∈ [n]. Since there are k j−1 vertices on each level, for a fixed j, the term will appear k j−1 times as an exponent in the denominator of our generating function.
When k = 2, the formulas for our exponents simplify nicely, as follows.
Corollary 3.5. For a binary tree with n levels, We omit the proof of the equivalence of the two rational forms in our Corollary, as they are the result of a tedious calculation. For those interested in the calculation, we mention that the key step is the use of

n-cycles
In this section we consider the open problem due to Corteel, Savage, and Wilf given in Example 1.5. Recall that the Laplacian minor for a general n-cycle on vertices 1, 2, . . . , n, labelled cyclically and minored at n, has the following form when rows and columns are indexed 1, 2, . . . , n − 1.

Define
K n := {λ ∈ R n−1 : L n,cyc λ ≥ 0} and σ Kn (z 1 , . . . , z n−1 ) := λ∈Kn∩Z n−1 Our goal is to study the integer point transform for K n by enumerating the integer points in the fundamental parallelepiped for K n ; along the way, we will see that there is a hidden additive structure to the ray generators of K n which is useful.
By Lemma 2.1, the rows of L n,cyc determine the normal vectors to the facets of K n , while the columns of L −1 n,cyc provide (non-integral) ray generators of K n . We first explicitly describe the entries in L −1 n,cyc .
Proof. Define the matrix B = (b i,j ) using the b-values given above. Let B · L n,cyc = (a i,j ); since L n,cyc and B are symmetric matrices, it follows that a i,j = a j,i . Entry a i,j is given by for x or y equal to n. It is straightforward to show that BL n,cyc = I through a case-by-case analysis.
As a representative example, for i < j we have We next scale the matrix L −1 n,cyc by n so that the columns are integral; in doing so, we find that there is group-theoretic structure to the ray generators of K n , in the following sense.
Proof. Let M = (m i,j ). Since m i,j is equal to either i(n − j) or j(n − i), it follows that m i,j ≡ −ij mod n. Consider column v 1 . We find v 1 = (−1 · 1, −2 · 1, . . . , −(n − 1) · 1) T mod n, Evaluating an arbitrary column Our next step is to use the columns of nL −1 n,cyc as ray generators for K n , leaving us only to understand the integer points in Π n , the fundamental parallelepiped for K n . For any integer point λ ∈ K n , since L n,cyc is an integer matrix, L n,cyc λ = c ∈ Z n−1 . Thus, every integer point  Proof. If x is an integer vector, then x 1 is certainly an integer. For the converse, suppose that x 1 is an integer. This happens precisely in the case where, for l denoting the first row of nL −1 n , we have that l·c = n j=1 l 1,j c j is an integer multiple of n.
Therefore, x k is also an integer.
Thus, the integer points in Π n are parametrized by the set of solutions to the system where c j ∈ {0, 1, 2, . . . , n − 1}. Equivalently, these points are parametrized by the set of partitions of multiples of n into positive parts not exceeding n − 1, with no more than n − 1 of each part. The solution set to this system of equations has arisen before in the study of "Hermite reciprocity" given in [15]. Our next theorem follows directly from the preceding discussion. .

While Theorem 4.4 and its specializations resolve the question of Corteel, Savage, and
Wilf, this resolution is not as clean as one might like. In the next section, we show how a slight modification of the graph underlying this problem leads to much more simply stated and elegant, but still closely related, univariate generating function identities.

Leafed n-cycles
In this section, we examine the case of compositions constrained by n-cycles augmented with a leaf; we will refer to such a graph as a leafed n-cycle. Throughout this section, we label the leaf vertex as n and label the vertices of the n-cycle cyclically as 0, 1, . . . , n − 1, where n is adjacent to 0 as depicted in Figure 1. For a leafed n-cycle, we denote by L n the minor of the Laplacian taken at vertex n and observe that it has the form where the columns (resp. rows) are labeled from top to bottom (resp. left to right) by 0, 1, 2, . . . , n − 1. Define C n := {λ ∈ R n : L n λ ≥ 0} and σ Cn (q) := λ∈Cn∩Z n q λ 0 .
While not immediately apparent, the reason for focusing on the first coordinate of integer points in C n will become clear.

A generating function identity.
Our main result regarding leafed cycles is the following generating function identity.
Theorem 5.1. Let S n denote the set of solutions to the system where c j ∈ {0, 1, 2, . . . , n − 1}. For c ∈ S n , define φ(c) := n−1 j=0 c j . Then Note that c∈Sn q φ(c) is the ordinary generating function for the "number of parts" statistic on the set of all non-negative partitions of multiples of n into parts not exceeding n − 1, with no more than n − 1 of each part. The similarity between Theorem 5.1 and Theorem 4.4 is immediate. However, by switching from n-cycles to leafed n-cycles, a more elegant univariate identity is obtained than those obtained by straightforward specializations of Theorem 4.4.
We remark that our proof of Theorem 5.1 can easily be extended to produce a multivariate statement analogous to Theorem 4.4; we omit this, since it is the specialization above that we find the most interesting consequence of the multivariate identity.
As in the previous section, our goal is to apply Lemma 2.3. We first explicitly describe the entries in L −1 n , which are closely related to the entries of L −1 n,cyc .
Proposition 5.2. For a leafed n-cycle and 0 ≤ i, j ≤ n − 1, Proof. Define the matrix B = (b i,j ) using the b-values given above. Let L n B = (a i,j ); since L n and B are symmetric matrices, it follows that a i,j = a j,i . For i = 0, entry a i,j is given by From these observations, it is straightforward to show that L n B = I through a case-by-case analysis.
For example, for j = 0, a 0,j = 0 is shown as follows.
Note that every entry in the top row (and in the first column, by symmetry) of L −1 n is equal to 1; it is this property of the inverse that allows us to effectively study the generating function recording only the first entry of each composition in C n . We next scale the matrix L −1 n by n so that the columns (i.e. ray generators of our cone) are integral; in doing so, we find that there is group-theoretic structure to the columns of L −1 n , again related closely to that for L −1 n,cyc .
Proof. Let M = (m i,j ). Since m i,j is equal to either i(n − j) + n or j(n − i) + n, it follows that m i,j ≡ −ij mod n. Consider column v 1 . We find v 1 = (−0 · 1, −1 · 1, −2 · 1, . . . , −(n − 1) · 1) T mod n, = (0, n − 1, n − 2, . . . , 1) T ∈ Z n n . 15 Evaluating an arbitrary column v k , 0 ≤ k ≤ n − 1, we have v k = (−0 · k, −1 · k, −2 · k, . . . , −(n − 1) · k) T mod n, As in the case of n-cycles, our next step is to use the columns of nL −1 n as ray generators for C n ; we must understand the integer points in Π n , which now denotes the fundamental parallelepiped for C n . For any integer point λ ∈ C n , since L n is an integer matrix, L n λ = c ∈ Z n . Thus, every integer point in C n can be expressed in the form L −1 n c for an integral c. As before, if λ ∈ Π n , then Proof. The proof is almost identical to that for the n-cycle case.
Thus, the integer points in Π n are parametrized by the set of solutions to the system where c j ∈ {0, 1, 2, . . . , n − 1}. Equivalently, these points are parametrized by the set of partitions of multiples of n into non-negative parts not exceeding n − 1, with no more than n − 1 of each part.
Our final observation in this subsection, and the key observation needed to apply Lemma 2.3 and complete the proof of Theorem 5.1, is that for any point λ such that L n λ = c ∈ Π n , we have that λ 0 = i c i . This is a consequence of Proposition 5.2, which shows that the first row (and column) of L −1 n are vectors of all ones.

5.2.
Cyclically distinct compositions. This subsection is devoted to the following conjecture, which provides a combinatorial interpretation for the coefficients of σ Cn (q).
Conjecture 5.5. The number of integer points λ ∈ C n with λ 0 = m is equal to the number of compositions of m into n parts that are cyclically distinct.
Conjecture 5.5 is based on experimental evidence for small values of n; further, we used LattE [18] to check the conjecture for all prime values of n up to n = 17. An obvious way to attack Conjecture 5.5 is to attempt to prove that for any λ ∈ C n with λ 0 = m, we have that λ = L −1 n c for an integer vector c whose entries form a weak composition of m that is cyclically distinct from all other such c vectors forming elements of C n . Unfortunately, this does not work out, as the following example demonstrates.
Example 5.6. The weak compositions of 3 into 3 parts are given by the following families     The four compositions which satisfy L −1 3 c ∈ C n are the elements of F 1 and F 4 . Thus, while this shows that Conjecture 5.5 holds in this case, it is not a consequence of the c-vectors being themselves cyclically distinct.
While this obvious approach to Conjecture 5.5 fails in certain cases, it is successful in others, as seen in the following theorem. Proof. We must show that for every composition c of m with n parts, exactly one of the cyclic shifts of c will produce an integer point when L −1 n is applied. Begin with a composition c = (c 0 , c 1 , . . . , c n−1 ) of m such that L −1 n c ∈ C n ∩ Z n . Then for all j, where v j = (v j,0 , v j,1 , . . . , v j,n−1 ) is the jth row of nL −1 n ( mod n). Now consider another cyclic ordering c ′ of c where we shift the position of each entry by k, so c ′ k = (c k , c k+1 , . . . , c k+n−1 ) with the indices taken mod n. Then the jth coordinate of L −1 n c ′ is given by By properties of v j given in Proposition 5.3, shifting the ith coordinate of v j by k is equivalent to subtracting the row v k from the row v j component-wise. So, Since n−1 i=0 c i ≡ 0 mod n by assumption, and since v j,k n−1 i=0 c i ≡ 0 mod n for j, k ≡ 0 due to n being prime, we have that Thus, the non-trivial cyclic shifts of c do not correspond to integer points in C n .
It is straightforward to show that for the case where n is prime and n|m, for any composition c of m into n parts that satisfies L −1 n c ∈ C n ∩ Z n all the cyclic shifts of c also satisfy this condition. Thus, the remaining challenge regarding Conjecture 5.5 in the case where n is prime is to show that this union of orbits of compositions under the cycle action contains as many compositions as there are orbits of the cycle action. While we have not been successful in proving this, the study of these compositions reveals a beautiful underlying geometric structure for our cone, which we discuss in the following section.

Reflexive polytopes and leafed cycles of prime length
For leafed cycles of prime length, the remaining cases of Conjecture 5.5 turn out to be equivalent to the problem of enumerating points in integer dilates of an integer polytope, which is the subject of Ehrhart theory. The polytope in question is a simplex, and we will prove that it has the additional structure of being reflexive, a property we now introduce. 6.1. Reflexive polytopes. An n-dimensional polytope P in R n may equivalently [25, Chapter 1] be described as the convex hull of at least n + 1 points in R n or as the intersection of at least n + 1 linear halfspaces in R n . When P = conv(V ) and V is the minimal set such that this holds, then we say the points in V are the vertices of P . If P = conv(V ) and |V | = n + 1, then we say P is a simplex. We say P is integral if the vertices of P are contained in Z n .
For P ⊂ R n an integral polytope of dimension n, and for t ∈ Z >0 , set tP := {tp : p ∈ P } and L P (t) := |Z n ∩ tP |, i.e. the number of integer points in tP is L P (t). Define the Ehrhart series for P to be Ehr P (x) := 1 + t≥1 L P (t)x t . A fundamental theorem due to E. Ehrhart [14] states that for an n-dimensional integral polytope P in R n , there exist complex values h * j so that A stronger result, originally due to R. Stanley in [23], is that the h * j 's are actually nonnegative integers; Stanley's proof of this used commutative algebra, though several combinatorial and geometric proofs have since appeared. We call the coefficient vector h * P := (h * 0 , . . . , h * d ) in the numerator of the rational generating function for Ehr P (x) the h-star vector for P . The volume of P can be recovered as ( j h * j )/n!. Obtaining a general understanding of the structure of h * -vectors of integral polytopes is currently of great interest. Recent activity, e.g. [2,5,7,9,17,16,20,22], has focused on the class of reflexive polytopes, which we now introduce as they will be needed in Section 6.
Given an n-dimensional polytope P , the polar or dual polytope to P is Let P • denote the topological interior of P . Definition 6.1. An n-dimensional polytope P is reflexive if 0 ∈ P • and both P and P ∆ are integral.
Reflexive polytopes have many rich properties, as seen in the following lemma. Lemma 6.2. [2,17] P is reflexive if and only if P is an integer polytope with 0 ∈ P • that satisfies one of the following (equivalent) conditions: (1) P ∆ is integral.
(2) L P • (t + 1) = L P (t) for all t ∈ Z ≥0 , i.e. all lattice points in R n sit on the boundary of some non-negative integral dilate of P .
where h * i is the i th coefficient in the numerator of the Ehrhart series for P .
Reflexive polytopes are simultaneously a very large class of integral polytopes and a very small one, in the following sense. Due to a theorem of Lagarias and Ziegler [19], there are only finitely many reflexive polytopes (up to unimodular equivalence) in each dimension.
On the other hand, Haase and Melnikov [16] proved that every integral polytope is a face of some reflexive polytope. Because of this "large yet small" tension, combined with their importance in other areas of mathematics as noted before, whenever a reflexive polytope arises unexpectedly it is a surprise, an indication that something interesting is happening.
6.2. Reflexive slices of Laplacian minor cones. Our main result in this subsection is Theorem 6.3, which asserts that for prime n values, reflexive polytopes arise as slices of the cone constrained by Laplacian minors for leafed n-cycles. Note that when n is not prime, this construction does not always yield reflexive simplices; this has been verified experimentally with LattE [18] and Lemma 6.2. However, exactly when the construction described below produces reflexive polytopes is at this time not clear.
Let p be an odd prime and let lCP p be the simplex obtained by intersecting the cone constrained by a leafed p-cycle with the hyperplane λ 1 = p in R p . Theorem 6.3. lCP p is reflexive (after translation by an integral vector).
Proof. By elementary results about polar polytopes found in [25,Chapter 2], P is a reflexive polytope if and only if P is integral, contains the origin in its interior, and has a half-space description of the form where A is an integral matrix and 1 denotes the vector of all ones.
We first observe that lCP p is clearly an integer polytope, as its vertices are the columns of pL −1 p , which are integral. A halfspace description of lCP p is given by λ 0 = p and Let w i be the ith column of L −1 p , and observe that L p ( i w i ) = L p ( i L −1 p (e i )) = 1. Thus, i w i is an interior integer point of our simplex lCP p ; we want to change coordinates so that i w i is translated to the origin. We therefore consider solutions to our halfspace description of the form λ + i w i . It follows that the description of our translated lCP p does not use a λ 0 -coordinate (since λ 0 + p = p implies λ 0 = 0), and is given by Since lCP p is an integral simplex, and i w i is an integer vector by which we translated lCP p , our translated lCP p is still integral. Thus lCP p is reflexive, and our proof is complete.
While the proof above is elementary, it ties the study of compositions constrained by graph Laplacian minors into an interesting circle of questions regarding reflexive polytopes. The only remaining case of Conjecture 5.5 is that the number of integer points in m · lCP p is equal to the number of compositions of mp with p parts, up to cyclic equivalence, which is now asking for a combinatorial interpretation for the Ehrhart series of a reflexive simplex.
Because of reflexivity, the generating function for this series yields a rational function with a symmetric numerator, which yields a functional relation on the Ehrhart polynomial for this simplex.
The study of reflexive polytopes goes hand in hand with the study of several other interesting classes of polytopes; one such example are normal polytopes, where an integer polytope P is normal if every integer point in the m-th dilate of P is a sum of exactly m integer points in P . In our attempts to prove the prime case of Conjecture 5.5, we noticed that our techniques (though unsuccessful at providing a proof) provided evidence suggesting that lCP p is normal. Normality is implied by the presence of a unimodular triangulation for an integral polytope; while we are not yet certain of the existence of such a triangulation for lCP p or of the normality of lCP p , it would not surprise us if such a triangulation can be found for all prime p.
Our final remark regarding lCP p , reflexivity, and normality, regards the unimodality of the h * vector of lCP p . It is a major open question (see [22] and the references therein) whether all normal reflexive polytopes have unimodal h * -vectors. For p ≤ 7, we found using LattE [18] that the h * -vector for lCP p is unimodal. We again suspect that this holds in general.

Leafed cycles of length a power of two
We conclude our paper by considering leafed cycles for non-prime values of n; specifically, we study when n = 2 k for some k. Using LattE [18], one observes that the cone constrained by a leafed 8-cycle does not have a reflexive 8-th slice (we suspect this to be the case for all non-prime values of n). Nevertheless, the integer point transform of this cone exhibits some interesting "near-symmetry" in the numerator for small powers of 2. Based on experimental evidence, we offer the following conjecture.
Conjecture 7.1. Let C 2 k be the cone constrained by the Laplacian minor of a leafed n-cycle where n = 2 k for some integer k ≥ 2. Then the generating function has the form σ C 2 k (q, 1, 1, . . . , 1) = f (q) where f (q) has the following property. Let (a 0 , . . . , a j ) denote the coefficient list of f (q). If we append a 0 to the end of this coefficient list, then take the difference between the appended coefficient list and its reverse, we obtain the coefficient list of the polynomial The near-symmetry of the numerator polynomial indicates that there should be some interesting structure to the finite group obtained by quotienting the semigroup of integer points in C 2 k by the semigroup generated by a specific choice of integral ray generators for the cone.