Counting Lattice Points in Minkowski Sums of Cross Polytopes

Abstract: Motivated by Postnikov's study of lattice-point enumeration in Minkowski sums of simplices, we investigate lattice points in Minkowski sums of cross polytopes and establish analogous results, together with several related consequences. In particular, we introduce the support-enumerator associated with Postnikov's notion of draconian sequences and show that it coincides with the $h<sup>*$-polynomial of the corresponding root polytope. This provides a new interpretation of the $h<sup>*$-polynomial and yields a simple method for computing the volume of the corresponding polytope. We further exploit the symmetry of such root polytopes to establish a duality property for support-enumerators, which in turn provides a proof of a conjecture by Chapoton and Athanasiadis concerning the $h$-polynomials of preorders. As an immediate consequence, we prove that a Minkowski sum of cross polytopes and its dual polytope have the same number of lattice points. This duality then leads to a general formula for the number of lattice points in Minkowski sums of cross polytopes in terms of draconian sequences. Our formula enables us to compute the Ehrhart polynomials of these polytopes and show that they are Ehrhart positive. Furthermore, the formula allows us to derive analogous formulas for the number of lattice points on their boundaries and for their surface volumes.

• The paper establishes a closed lattice-point formula, summing products of coefficients c(a_i,k_i) over graph-defined draconian sequences for arbitrary nonnegative parameters.
• It identifies the support-enumerator F_G(z) with the h*-polynomial of an augmented bipartite graph’s root polytope, yielding normalized volume |D(G)| and proving a duality conjecture.
• It proves strict Ehrhart positivity for every such cross-polytope sum and derives explicit formulas for interior points, boundary points, and surface volume, while leaving unimodality and real-rootedness open.

Overview and motivation

This paper develops a lattice-point enumeration theory for polytopes of the form $$Q_G(k_1,\dots,k_m) = k_1\diamondsuit_{I_1} + \cdots + k_m\diamondsuit_{I_m}$$, where $$\diamondsuit_I = \mathrm{conv}(\pm e_i \mid i \in I)$$ is a cross polytope, and $G=(L\sqcup R,E)$ is a bipartite graph whose left vertex $\ell_j$ has neighborhood $I_j \subseteq [n]$. The work is directly modeled on Postnikov's enumeration of Minkowski sums of simplices $P_G(y_1,\dots,y_m)$ via draconian sequences [Postnikov2009], and the paper positions itself as the type B counterpart to that theory. The family $Q_G$ consists of type B generalized permutohedra (a subfamily in the sense of Bastidas [Bastidas2021] and Eur–Fink–Larson–Spink [Euretall2024]), so general formulas for lattice points already exist; the paper's stated advantage is that its formula makes Ehrhart positivity transparent, which the existing type B formulas do not.

Basic geometry of cross-polytope sums

Two structural facts anchor the analysis. First, $\dim Q_G(k_1,\dots,k_m) = |U|$, where $U = I_1\cup\cdots\cup I_m$ is the set of non-isolated right vertices. Second, and more importantly, the intersection of $Q_G(k_1,\dots,k_m)$ with the nonnegative orthant is exactly $$\diamondsuit_I = \mathrm{conv}(\pm e_i \mid i \in I)$$0: each orthant section is a reflection of this simplex sum. Consequently, $$\diamondsuit_I = \mathrm{conv}(\pm e_i \mid i \in I)$$1 admits the inequality description

$$\diamondsuit_I = \mathrm{conv}(\pm e_i \mid i \in I)$$2

for all nonempty $$\diamondsuit_I = \mathrm{conv}(\pm e_i \mid i \in I)$$3, and its volume equals $$\diamondsuit_I = \mathrm{conv}(\pm e_i \mid i \in I)$$4 times the volume of the corresponding simplex sum, giving

$$\diamondsuit_I = \mathrm{conv}(\pm e_i \mid i \in I)$$5

where $$\diamondsuit_I = \mathrm{conv}(\pm e_i \mid i \in I)$$6 denotes the set of $$\diamondsuit_I = \mathrm{conv}(\pm e_i \mid i \in I)$$7-draconian sequences $$\diamondsuit_I = \mathrm{conv}(\pm e_i \mid i \in I)$$8 satisfying $$\diamondsuit_I = \mathrm{conv}(\pm e_i \mid i \in I)$$9 for every nonempty $G=(L\sqcup R,E)$0.

Support-enumerators as $G=(L\sqcup R,E)$1-polynomials of root polytopes

The key new combinatorial object is the support-enumerator

$G=(L\sqcup R,E)$2

where $G=(L\sqcup R,E)$3 counts positive coordinates. The main structural theorem states that $G=(L\sqcup R,E)$4 is the $G=(L\sqcup R,E)$5-polynomial of the root polytope $G=(L\sqcup R,E)$6 of the augmented bipartite graph $G=(L\sqcup R,E)$7:

$G=(L\sqcup R,E)$8

The proof slices $G=(L\sqcup R,E)$9 by fixing integral left marginals $\ell_j$0 with $\ell_j$1, showing each slice is integrally equivalent to a simplex sum $\ell_j$2, and then applying Postnikov's counting formula together with a telescoping computation over the free coordinate $\ell_j$3.

This gives a new combinatorial interpretation of the $\ell_j$4-polynomial of these root polytopes — complementing Kálmán–Postnikov's interior polynomial interpretation [KalmanPostnikov2017] — and yields the immediate corollary

$\ell_j$5

a simple route to normalized volumes and Ehrhart polynomials of root polytopes that bypasses the spanning-tree activity statistics of Kálmán–Tóthmérész [KalmanTothmeresz2023].

Duality and the Chapoton–Athanasiadis conjecture

Since $\ell_j$6 under coordinate swap, the root polytopes $\ell_j$7 and $\ell_j$8 are integrally equivalent, forcing $\ell_j$9. This duality has three consequences:

• It resolves Conjecture 5.4 of Chapoton and Athanasiadis [chapoton2026] concerning the duality of $I_j \subseteq [n]$0-polynomials of preorders.
• It implies that $I_j \subseteq [n]$1 and its "dual" $I_j \subseteq [n]$2 have the same number of lattice points — indeed the same distribution of nonzero coordinates, since $I_j \subseteq [n]$3.
• It enables the central counting formula below.

The counting formula and Ehrhart positivity

Introducing clone graphs $I_j \subseteq [n]$4 (each $I_j \subseteq [n]$5 replaced by $I_j \subseteq [n]$6 clones with neighborhood $I_j \subseteq [n]$7), so that $I_j \subseteq [n]$8, the paper proves:

Main formula. For all $I_j \subseteq [n]$9,

$P_G(y_1,\dots,y_m)$0

where $P_G(y_1,\dots,y_m)$1 counts integer vectors in $P_G(y_1,\dots,y_m)$2 with coordinate $P_G(y_1,\dots,y_m)$3-norm equal to $P_G(y_1,\dots,y_m)$4:

$P_G(y_1,\dots,y_m)$5

The proof groups draconian sequences of the clone graph by group-sums, using the fact that group-sums of $P_G(y_1,\dots,y_m)$6-draconian sequences are exactly $P_G(y_1,\dots,y_m)$7.

Three properties of $P_G(y_1,\dots,y_m)$8 drive the remaining consequences: it is a degree-$P_G(y_1,\dots,y_m)$9 polynomial in $Q_G$0 with leading coefficient $Q_G$1, it has only nonnegative rational coefficients, and its monomials have parity matching $Q_G$2 (equivalently, $Q_G$3). These are established via the generating function identity $Q_G$4.

Ehrhart positivity. Since $Q_G$5, the Ehrhart polynomial is $Q_G$6, and nonnegativity of the coefficients of each factor gives nonnegative coefficients throughout; strict positivity of every coefficient up to degree $Q_G$7 follows by constructing, for each $Q_G$8, a draconian sequence of total weight $Q_G$9 supported on indices with $\dim Q_G(k_1,\dots,k_m) = |U|$0. This is a strong claim: every Minkowski sum of cross polytopes indexed by an arbitrary bipartite graph is strictly Ehrhart positive, and the positivity is manifest from the formula — a contrast the authors draw explicitly with the signed-simplex decompositions required when expressing cross polytopes via existing type B formulas, where negative Minkowski coefficients obscure positivity.

Interior, boundary, and surface volume

The parity property $\dim Q_G(k_1,\dots,k_m) = |U|$1 allows Ehrhart reciprocity to be applied termwise. Splitting $\dim Q_G(k_1,\dots,k_m) = |U|$2 into even- and odd-weight parts with support-enumerators $\dim Q_G(k_1,\dots,k_m) = |U|$3 and $\dim Q_G(k_1,\dots,k_m) = |U|$4, the paper obtains explicit formulas for the number of interior lattice points, $\dim Q_G(k_1,\dots,k_m) = |U|$5 at dilation one, and for boundary points ($\dim Q_G(k_1,\dots,k_m) = |U|$6 or $\dim Q_G(k_1,\dots,k_m) = |U|$7 according to parity of $\dim Q_G(k_1,\dots,k_m) = |U|$8). The surface-volume formula follows from extracting the coefficient of $\dim Q_G(k_1,\dots,k_m) = |U|$9: since no sequence with $U = I_1\cup\cdots\cup I_m$0 or $U = I_1\cup\cdots\cup I_m$1 contributes to that coefficient,

$U = I_1\cup\cdots\cup I_m$2

Limitations and open questions

Several caveats are stated plainly. The counting formula applies directly to nonnegative dilations; validity under negative $U = I_1\cup\cdots\cup I_m$3 requires the Minkowski-difference condition $U = I_1\cup\cdots\cup I_m$4, inherited from Bastidas's results on type B generalized permutohedra — it is not automatic. The paper also concedes that not all properties conjectured by Chapoton–Athanasiadis hold generally: palindromicity fails for support-enumerators in general, so the natural question of which bipartite graphs yield unimodal, palindromic, or real-rooted $U = I_1\cup\cdots\cup I_m$5 remains open. Further open problems include extending the slicing technique to $U = I_1\cup\cdots\cup I_m$6-polynomials of general Cayley polytopes (of which root polytopes are a special case), extending the preorder techniques of Chapoton–Athanasiadis to the full family, and developing face-enumeration ($U = I_1\cup\cdots\cup I_m$7- and $U = I_1\cup\cdots\cup I_m$8-vectors) for cross-polytope sums analogous to Postnikov–Reiner–Williams's treatment of generalized permutohedra [PostnikovEtal2008]. No log-concavity or unimodality result for the Ehrhart coefficients themselves is claimed.

Conclusion

The paper transfers Postnikov's draconian-sequence machinery from simplex sums to cross-polytope sums, proving a closed counting formula, strict Ehrhart positivity with manifest positivity, boundary and surface-volume refinements, and a new $U = I_1\cup\cdots\cup I_m$9-polynomial interpretation of root polytopes that settles a duality conjecture of Chapoton–Athanasiadis. The central technical contribution — identifying support-enumerators with $Q_G(k_1,\dots,k_m)$0-polynomials of augmented-graph root polytopes — both supplies the duality and provides an efficient volume computation, and the enumerated open questions delineate the extent to which the parallel between types A and B remains incomplete.