Polytopes

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#Polytopes

A polytope is called a lattice polytope or integral polytope if all its vertices have integer coordinates. If its vertices have rational coordinates, we call it a rational polytope. A convex polytope is called a regular polytope if its Euclidean symmetry group acts transitively on its flags, that is, on maximal chains of faces.

M.-C. Brandenburg, J. A. D. Loera, and C. Meroni study exact algorithms for cutting polytopal measures, meaning Lebesgue measure restricted to finite unions of full-dimensional convex polytopes [BLM26]. For fixed ambient dimension, their algorithms describe all ham-sandwich cuts and more general prescribed-proportion cuts as semialgebraic sets in polynomial time. They also compute centerpoints of rational polytopal measures, and show that the centerpoints of a convex polytope are exactly its floating body at level $1/(d+1).$

#Face $h$- and toric $g$-polynomials

Let $P$ be a $d$-dimensional simplicial convex polytope, and let $f_i$ be the number of faces of its boundary with $i$ vertices, including the empty face with $f_0=1.$ The face $h$-polynomial is defined by \[h(P;t)=\sum_{i=0}^{d} f_i t^i(1-t)^{d-i}.\] Its coefficient vector is the face $h$-vector of $P.$ We include the word face here to distinguish this polynomial from the $h^*$-polynomial below.

Theorem (Guo–Kang, [Thm. 1.1, GK26]).

Let \[h(t)=h_0+h_1t+\dotsb+h_dt^d\in\setZ_{\geq 0}[t]\] be monic, palindromic, and real-rooted. Then there is a $d$-dimensional simplicial convex polytope $P$ such that $h(P;t)=h(t).$

The converse does not hold: the face $h$-polynomial of a $d$-simplex is $1+t+\dotsb+t^d,$ which is not real-rooted for $d\geq2.$ Guo and Kang prove the theorem by showing that the gamma-vector of $h$ is an O-sequence, turning this into the required $g$-vector, and applying the $g$-theorem.

For an $n$-dimensional simple polytope $P,$ we define $h(P;x)$ as the face $h$-polynomial of its polar simplicial polytope. Write \[h(P;x)=\sum_{j=0}^{\lfloor n/2\rfloor} \gamma_jx^j(1+x)^{n-2j}.\] The vector $(\gamma_0,\dotsc,\gamma_{\lfloor n/2\rfloor})$ is the gamma-vector of $P.$ For $n\geq j,$ define the toric $g$-contribution polynomial by \[g_{n,j}(x)= \sum_{k=0}^{\min(\lfloor n/2\rfloor,n-j)} C_{n-k-j}\binom{n-k}{k}(x-1)^k, \qquad C_m=\frac{1}{m+1}\binom{2m}{m}.\] R. Ehrenborg, G. Hetyei, and M. Readdy proved that the toric $g$-polynomial of $P$ has the expansion \[g(P;x)=\sum_{j=0}^{\lfloor n/2\rfloor}\gamma_jg_{n,j}(x)\] [Thm. 3.4, EHR26].

Theorem (Xiao, [Thm. 1.3, Xia26]).

For $n\geq0$ and $0\leq j\leq\lfloor n/2\rfloor,$ the polynomial $g_{n,j}(x)$ is real-rooted and interlaces $g_{n+1,j}(x).$ If also $j\leq\lfloor(n-1)/2\rfloor,$ then $g_{n,j}(x)$ interlaces $g_{n+1,j+1}(x).$

The theorem proves the real-rootedness conjecture of Ehrenborg, Hetyei, and Readdy. Alexandersson subsequently proved Xiao’s row-wise strengthening.

Theorem (Alexandersson, [Thm. 8.1, Ale26]).

Let $n\geq2$ and $m=\lfloor n/2\rfloor.$ Then $g_{n,m}(x)/x$ interlaces every $g_{n,j}(x),$ and the sequence \[\bigl(g_{n,0}(x),g_{n,1}(x),\dotsc,g_{n,m}(x)\bigr)\] is an interlacing sequence. Consequently, every nonzero nonnegative linear combination of these polynomials is real-rooted.

In particular, every simple polytope with a nonnegative gamma-vector has a real-rooted toric $g$-polynomial. The same theorem gives real-rooted weak-ascent and strict-descent enumerators for weakly $123$-avoiding parking functions [Cor. 8.4, Ale26].

#The Ehrhart polynomial and $h^*$-vector

Let $P$ be a lattice polytope. The number of lattice points in the dilation $n\cdot P$ is a polynomial in $n.$ The Ehrhart polynomial of $P$ is the polynomial $E_P(n) \coloneqq |n P \cap \setZ^d|.$ The Ehrhart series is \[\sum_{n \geq 0} |nP \cap \setZ^d| t^n = \frac{H_P(t)}{(1-t)^{d+1}}\] where $H_P(t) = h^*_0 + h^*_1 t + \dotsb + h^*_d t^d$ is the $h^*$-polynomial of $P.$ The vector $(h^*_0 , h^*_1, \dotsc, h^*_d )$ is the $h^*$-vector; all its entries are nonnegative integers, see [Sta93] and [BS07]. For inequalities that every $h^*$-vector must satisfy, see the $h^*$-inequalities subsection on the Ehrhart theory page, which gives a more focused overview of this material.

For a $d$-dimensional polytope, we have the following relation between entries in the $h^*$-vector and the Ehrhart polynomial: \[E_P(t) = \sum_{j=0}^d \binom{t+d-j}{d} h^*_j.\] There is a similar relation to express $H_P(t)$ in terms of the Ehrhart polynomial coefficients involving the Eulerian polynomials.

V. Reiner and B. Rhoades introduce a $q$-deformation of the Ehrhart series using harmonic spaces and Macaulay inverse systems for finite point configurations [RR24]. They conjecture that this $q$-Ehrhart series is rational and construct a bigraded algebra whose Hilbert series recovers the same series.

For a rational polytope $P$ with denominator $q$ (the smallest positive integer such that $qP$ is a lattice polytope), the lattice-point counting function $E_P(t) = |tP \cap \setZ^d|$ is a quasi-polynomial of degree $d$ and period dividing $q.$ The Ehrhart series then satisfies \[\sum_{n \geq 0} E_P(n) z^n = \frac{h^*(P;z)}{(1-z^q)^{d+1}},\] where $h^*(P;z)$ is a polynomial of degree at most $q(d+1)-1$ with nonnegative integer coefficients, see [BS07] and [Sta93].

J. A. D. Loera, L. Escobar, N. Kaplan, and C. Wang develop a weight-lifting construction for sums of weighted lattice points of polytopes [LEKW24]. For many quasi-polynomial weights, the weighted sum over lattice points in $P$ is transformed into an ordinary lattice-point count in a higher-dimensional polytope. Their applications include representation-theoretic multiplicities such as Kostka coefficients.

E. Bajo, R. Davis, J. A. D. Loera, A. Garber, S. G. Mora, K. Jochemko, and J. Yu develop a weighted Ehrhart theory extending Stanley’s nonnegativity theorem [BDDL+24]. They prove nonnegativity for weighted $h^*$-polynomials when the weights are homogeneous polynomials decomposable into products of linear forms which are nonnegative on the polytope.

Some rational polytopes exhibit period collapse, meaning the Ehrhart quasi-polynomial is in fact a polynomial. Certain Gelfand–Tsetlin polytopes have this property. However, not every such polynomial arises as the Ehrhart polynomial of a lattice polytope. For example, [HM08] construct a rational polygon $Q$ whose Ehrhart function is a polynomial with linear coefficient $1,$ but no integral polygon has this property.

D. Anderson and A. Shah introduce a weighted $q$-enumeration of lattice points in a polytope [AS23]. They prove a Brion-type formula for the corresponding generating function and relate the construction to toric arc schemes of normal fans.

#Ehrhart–Macdonald reciprocity

Let $P$ be a $d$-dimensional rational polytope with Ehrhart polynomial $E_P(t).$ The Ehrhart–Macdonald reciprocity states that \[E_P(-t) = (-1)^d E_{P^\circ}(t)\] where $E_{P^\circ}(t) = |t P^\circ \cap \setZ^d|$ counts the lattice points in the interior of $tP.$ In terms of the Ehrhart series, this is equivalent to the relation $H_P(t) = t^{d+1} H_P(1/t),$ meaning the $h^*$-vector is palindromic if and only if the polytope is Gorenstein. The reciprocity can be seen as a discrete analogue of Poincaré duality. This was first proved by I. G. Macdonald [Mac71] for lattice polytopes; see [BS07] for an alternative proof in the rational case.

#Stanley’s monotonicity theorem

Theorem (Stanley Monotonicity, [Sta93]).

Suppose that $P \subseteq Q$ are rational polytopes with $qP$ and $qQ$ integral (for minimal possible $q \in \setZ_{\gt{}0}$). Define the $h^*$-polynomials via \[\operatorname{Ehr}(P;z) = \frac{h^*(P;z)}{(1-z^q)^{\dim(P)+1}} \qquad \text{and} \qquad \operatorname{Ehr}(Q;z) = \frac{h^*(Q;z)}{(1-z^q)^{\dim(Q)+1}}.\] Then $h^*(P;z) \leq h^*(Q;z)$ coefficient-wise.

See [BS07] for an alternative proof.

#Families of polytopes

#Hypersimplices

For $0\lt{}k\lt{}n,$ the hypersimplex $\Delta_{k,n}$ is \[\Delta_{k,n} \coloneqq \left\{ x\in[0,1]^n : x_1+x_2+\dotsb+x_n=k \right\}.\] It is the base polytope of the uniform matroid $U_{k,n},$ and hence a generalized permutohedron. Its normalized volume is the Eulerian number $A(n-1,k-1).$

N. Li proved a conjecture of R. P. Stanley giving a descent–exceedance interpretation of the $h^*$-polynomial of hypersimplices [Li12]. More precisely, for the half-open hypersimplex \[\widetilde{\Delta}_{k,n} \coloneqq \left\{ x\in[0,1]^{n-1} : k-1 \lt{} x_1+x_2+\dotsb+x_{n-1} \leq k \right\},\] one has \[h^*(\widetilde{\Delta}_{k,n};t) = \sum_{\substack{w\in\symS_{n-1}\\ \operatorname{exc}(w)=k-1}} t^{\operatorname{des}(w)}.\] The proof uses a shelling of a unimodular triangulation of the hypersimplex. A second shelling gives another formula using descents and Li’s cover statistic, and the construction extends to generalized hypersimplices related to algebras of Veronese type [Thms. 1.2, 1.3 and 7.3, Li12].

#Order polytopes

Given a partial order $P$ on the set $[n],$ we associate a polytope $\mathcal{O}(P),$ called the order polytope of $P.$ It is the polytope in $\setR^n$ defined via the inequalities \[\begin{aligned} x_i \leq x_j \text{ if } i \lt{}_P j \qquad \text{and} \qquad 0 \leq x_i \leq 1 \text{ for } 1\leq i \leq n. \end{aligned}\]

There are order polytopes with negative Ehrhart coefficients, see [Fig. 3.87, Sta11] and [Ale19].

Example (The smallest-dimensional negative order-polytope example).

Let $A_7$ be a seven-element antichain and let $P_{7,7}=A_7\oplus A_7$ be the ordinal sum. Thus $P_{7,7}$ has seven minimal elements and seven maximal elements, with every minimal element below every maximal element. Its order polytope has dimension $14$ and \[E_{\mathcal O(P_{7,7})}(k) =1-\frac{3041}{1430}k+\frac{18397}{4290}k^2+\dotsb,\] so it is not Ehrhart positive [Ex. 3.2 and Table 1, LT19].

This dimension is sharp: every order polytope of dimension at most $13$ is Ehrhart positive [Thm. 3.1, LXZ26]. In contrast, its $h^*$-polynomial is \[h^*_{\mathcal O(P_{7,7})}(t)=A_7(t)^2,\] where $A_7(t)$ is the Eulerian polynomial. Thus the $h^*$-coefficients are nonnegative even though the ordinary Ehrhart polynomial has a negative coefficient.

Let $Z_n$ be the zig-zag poset with cover relations $z_1\lt{}z_2\gt{}z_3\lt{}z_4\gt{}\dotsb.$ The $h^*$-polynomial of the order polytope $\mathcal{O}(Z_n)$ has a combinatorial interpretation in terms of the swap statistic on alternating permutations. This also gives another proof that its coefficients are symmetric and unimodal; see [CS23]. T. Lundström and L. S. M. Leite study the order polytopes of crown posets [LSML26]. They give formulas for $f$-vectors, recurrences for Ehrhart polynomials, and an interpretation of the $h^*$-polynomial using cyclically alternating permutations.

B. Braun and A. Jal prove that order polytopes of generalized snake posets have real-rooted $h^*$-polynomials [BJ26]. These posets are width-two distributive lattices, and the proof uses Stanley’s interpretation of the $h^*$-polynomial as a $P$-Eulerian polynomial, its realization as a non-nesting rook polynomial, and interlacing recurrences. The ladder and regular snake cases recover the Narayana and Delannoy polynomials, respectively.

#Chain polytopes

Given a finite poset $P,$ its chain polytope $\mathcal{C}(P)$ is the polytope in $\setR_{\geq 0}^{P}$ cut out by \[x_{p_1}+x_{p_2}+\dotsb+x_{p_k}\leq 1\] for every chain $p_1\lt{}_P p_2\lt{}_P\dotsb\lt{}_P p_k.$ Stanley introduced chain polytopes together with order polytopes [Sta86]; the two have the same Ehrhart polynomial.

#Marked order polytopes

Let $P$ be a finite poset, let $A\subseteq P,$ and let $\lambda:A\to\setZ$ be order preserving. The marked order polytope is \[\mathcal O_{P,A}(\lambda) = \left\{ \widehat\lambda:P\to\setR: \widehat\lambda\text{ is order preserving and } \widehat\lambda|_A=\lambda \right\}.\] When $A$ contains all minimal and maximal elements of $P,$ this is a bounded lattice polytope. It generalizes the order polytope; see [ABS11].

Theorem (Jochemko–Menon [Thms. 1.3–1.5, JM26]).

Suppose $P$ belongs to a family $\mathcal F$ of posets closed under taking ideals and filters, and the linear coefficient of the order polynomial of every poset in $\mathcal F$ is nonnegative. Then $\mathcal O_{P,A}(\lambda)$ is Ehrhart positive whenever $A\supseteq\min(P)\cup\max(P).$

More strongly, on a chamber with $\lambda(a_0)\leq\dotsb\leq\lambda(a_r)$ for a natural labeling $A=\{a_0,\dotsc,a_r\},$ its lattice-point count is a polynomial with nonnegative coefficients in the gaps $t_i=\lambda(a_i)-\lambda(a_{i-1}).$ In particular, this applies to every integrally marked order polytope whose underlying poset is the cell poset of a skew shape.

The source of the positivity is an ideal-chain decomposition [Prop. 2.1, JM26]. For positive gaps, the lattice points split, over strict chains of order ideals $I_0\subsetneq\dotsb\subsetneq I_r$ with $a_i\in I_i\setminus I_{i-1},$ into independent lattice points of unmarked order polytopes. Equivalently, with $I_{-1}=\varnothing,$ \[\Omega_{P,A}(\lambda) = \sum_I \prod_{i=1}^r \Omega_{I_i\setminus(I_{i-1}\cup\mathord\uparrow a_i)}(t_i).\] The closure hypothesis keeps every poset in these factors inside $\mathcal F$; positivity of their order polynomials then makes every summand coefficientwise nonnegative.

J. Stricker studies $2$-levelness for marked order, marked chain, and marked chain-order polytopes, and gives an exact formula for the Ehrhart polynomial of marked order polytopes [Str24]. Via transfer between the marked models, the same polynomial also gives the Ehrhart polynomial for the corresponding marked chain and marked chain-order polytopes.

The Gelfand–Tsetlin polytopes are marked order polytopes.

#Cross-polytopes

The cross-polytope in dimension $d$ is \[\left\{ x\in\setR^d : |x_1|+\dotsb+|x_d|\leq 1 \right\}.\] K. Menon and E. Verkama give a combinatorial interpretation for scaled Ehrhart coefficients of cross-polytopes, and more generally for pyramids over cross-polytopes, in terms of colored permutations [MV25]. In particular, this gives a combinatorial proof that these Ehrhart coefficients are positive.

#Root polytopes

For a finite root system $\Phi\subseteq\setR^d,$ the root polytope is the convex hull $\operatorname{conv}(\Phi).$ In type $A,$ one often also considers graphical root polytopes: for a graph $G$ on $[n]$ with oriented edges $i\to j,$ the associated root polytope is \[Q_G\coloneqq \operatorname{conv}\left(\{0\}\cup\{e_i-e_j : i\to j \text{ is an edge of }G\}\right).\]

#Flow polytopes

Let $G=(V,E)$ be a directed graph and let $a\in\setR^V$ satisfy \[\sum_{v\in V} a_v=0.\] The flow polytope $\mathcal{F}_G(a)$ is the set of nonnegative flows $f:E\to\setR_{\geq 0}$ such that, for every vertex $v,$ \[\sum_{e=(v,w)} f(e)-\sum_{e=(u,v)} f(e)=a_v.\] Thus the vector $a$ records the netflow at each vertex. Flow polytopes are closely related to root polytopes, Kostant partition functions, and Gelfand–Tsetlin polytopes; see for example [LMD19]. R. S. González D’León, C. R. H. Hanusa, and M. Yip introduce permutation flows as a combinatorial model for triangulations of flow polytopes [DHY25]. Their construction gives a recursive triangulation model in which top-dimensional simplices are indexed by permutation flows.

#The permutohedron

The permutohedron is the polytope \[\operatorname{Perm}_n \coloneqq \operatorname{conv} \left\{ (w(1),w(2),\dotsc,w(n)) : w\in\symS_n \right\}.\] Equivalently, it is the convex hull of all coordinate permutations of $(1,2,\dotsc,n).$ Its face structure is governed by ordered set partitions of $[n],$ and it is a basic example of a generalized permutohedron; see [Zie95]. Its $h$-polynomial is the Eulerian polynomial $A_n(t).$

#Generalized permutohedra

A generalized permutohedron is a polytope whose normal fan coarsens the braid fan. Equivalently, every edge is parallel to $e_i-e_j$ for some $i,j.$ Many generalized permutohedra can be described by submodular functions; in particular, matroid base polytopes are generalized permutohedra.

The biEulerian polynomial $B_n(t)$ counts bipermutations by descents and is the $h$-polynomial of the bipermutahedral fan. It is real-rooted [Thm. 6.3, Ard22]; hence this fan has a log-concave and unimodal $h$-vector.

#Birkhoff polytopes

The Birkhoff polytope $\mathcal{B}_n$ is the polytope of $n\times n$ doubly stochastic matrices: \[\mathcal{B}_n \coloneqq \left\{ X\in\setR_{\geq 0}^{n\times n} : \sum_j X_{ij}=1 \text{ for all } i,\, \sum_i X_{ij}=1 \text{ for all } j \right\}.\] Equivalently, by the Birkhoff–von Neumann theorem, $\mathcal{B}_n$ is the convex hull of the $n\times n$ permutation matrices. See [Pak00] for several combinatorial questions about this polytope. The Birkhoff polytope also appears as a Gelfand–Tsetlin polytope.

E. Banaian, S. Chepuri, E. Gunawan, and J. Pan define $c$-Birkhoff polytopes by taking convex hulls of permutation matrices indexed by $c$-singletons for a Coxeter element $c$ [BCGP25]. In type $A,$ these polytopes are integrally equivalent to order polytopes of heap posets. Their type $B$ version embeds signed permutations in $B_n$ into $S_{2n}$ and gives the same conclusion: for every Coxeter element $c_B$ of $B_n,$ the corresponding type $B$ $c$-Birkhoff polytope is integrally equivalent to the order polytope of the heap of the longest $c_B$-sorting word [Thm. 3.16, BCGP26].

#Edge polytopes

Let $G=([n],E)$ be a finite simple graph. The edge polytope of $G$ is \[P_G \coloneqq \operatorname{conv}\{e_i+e_j : \{i,j\}\in E\} \subseteq \setR^n.\] The related symmetric edge polytope is \[P_G^{\pm} \coloneqq \operatorname{conv}\{\pm(e_i-e_j): \{i,j\}\in E\}.\] Symmetric edge polytopes often appear in Ehrhart-theoretic questions, for example in gamma-positivity and magic positivity problems.

Theorem (Real-rooted symmetric-edge $h^*$-polynomials).

For $a,b\geq0,$ let \[h^*_{a,b}(t)=h^*(P_{K_{a+1,b+1}}^{\pm},t).\] Then $h^*_{a,b}(t)$ is real-rooted, and for $b\geq1,$ \[h^*_{a,b-1}(t)\interl h^*_{a,b}(t)\] [Thm. 4.8, HJM19].

If $G$ is a cactus graph and $\widehat G$ is its suspension, then $h^*(P_{\widehat G}^{\pm},t)$ is real-rooted [Thm. 1.2, OT21].

The cactus result uses matching-generating polynomials, linking it to the matching-polynomial method.

#Perfect matching polytopes

For a graph $G=(V,E),$ the perfect matching polytope is the convex hull in $\setR^E$ of the indicator vectors of perfect matchings of $G.$ For the complete bipartite graph $K_{n,n},$ this is the Birkhoff polytope.

#Associahedra

The associahedron, also called the Stasheff polytope, is the polytope whose vertices correspond to triangulations of a convex polygon and whose edges correspond to diagonal flips. Equivalently, its face poset records partial triangulations of the polygon.

#Parking function polytopes

Let $u=(u_1,\dotsc,u_n)\in\setR_{\geq 0}^n$ satisfy $u_1\leq\dotsb\leq u_n.$ A vector $a=(a_1,\dotsc,a_n)\in\setR_{\geq 0}^n$ is a $u$-parking function if its non-decreasing rearrangement $b_1\leq\dotsb\leq b_n$ satisfies $b_i\leq u_i$ for all $i.$ The parking function polytope $\operatorname{PF}(u)$ is the convex hull of all $u$-parking functions.

F. Liu and W. Thawinrak study these polytopes, including their normal fans, face posets, $h$-polynomials, volumes, and Ehrhart polynomials [LT25]. In this nonnegative convention, the classical parking functions correspond to $u=(0,1,\dotsc,n-1).$

M. Hanada, J. Lentfer, and Andrés R. Vindas-Meléndez study the subfamily coming from $\xvec=(a,b,\dotsc,b)$ [HLV23]. They relate these parking-function polytopes to Pitman–Stanley polytopes and partial permutahedra, and obtain a closed formula for the volume.

#The integer decomposition property

An integral polytope $\mathcal{P} \subset \setR^d$ is said to have the integer decomposition property (IDP) if for every positive integer $k$ and $\xvec \in k \mathcal{P} \cap \setZ^d,$ we can find $\xvec_1,\xvec_2,\dotsc,\xvec_k \in \mathcal{P} \cap \setZ^d$ such that $\xvec_1 + \dotsb + \xvec_k = \xvec.$

In the following list, each entry implies the next:

$\mathcal{P}$ has a unimodular triangulation.
$\mathcal{P}$ has the integer decomposition property.
$\mathcal{P}$ is integral.

Using Cayley sums of rectangular prisms, L. Ferroni constructs smooth IDP polytopes with non-unimodal $h^*$-polynomials, disproving the unimodality conjecture attributed to Stanley [Fer26]. The same construction also gives Gorenstein IDP polytopes with non-log-concave $h^*$-polynomials and IDP polytopes whose Ehrhart series is not log-concave, disproving conjectures of Brenti and of Ferroni–Higashitani, respectively.

#Reflexive and Gorenstein polytopes

A lattice polytope $P \subset \setR^d$ containing the origin in its interior is called reflexive if the dual polytope \[P^* \coloneqq \left\{ y \in \setR^d : \langle x, y \rangle \leq 1 \text{ for all } x \in P \right\}\] is again a lattice polytope. Equivalently, $h^*_P(t)$ is palindromic of degree $d,$ see [Hib92].

A polytope $P$ is called Gorenstein (of index $r$) if for some $r\in\setZ_{\gt{}0},$ the dilation $rP$ contains a unique lattice point $m,$ and the translation $rP-m$ is reflexive.

#Magic positivity and reflexive polytopes

Following [FH24], a degree-$d$ polynomial $f(n)$ is called magic positive if it can be written in the form \[f(n) = \sum_{i=0}^d a_i n^i (n+1)^{d-i}\] with $a_i \geq 0$ for all $i.$ For Ehrhart polynomials, this basis is useful because it links coefficient positivity to real-rootedness of the corresponding $h^*$-polynomial.

Theorem (Brändén; see [Thm. 4.19, FH24]).

If the Ehrhart polynomial $E_P(n)$ of a lattice polytope is magic positive, then $E_P(n)$ has nonnegative coefficients and $h^*_P(t)$ is real-rooted.

If $W$ denotes Wagner’s transformation characterized by \[\sum_{n \geq 0} p(n) z^n = \frac{W(p)(z)}{(1-z)^{1+\deg p}},\] then magic positivity implies that $W(p)$ is real-rooted, see [Rem. 4.21, FH24]. For Ehrhart polynomials, $W(E_P)$ is exactly the $h^*$-polynomial. This is the same transformation discussed on the page about interlacing polynomials.

Cartesian products satisfy $E_{P\times Q}(n)=E_P(n)E_Q(n).$ Consequently, if the $h^*$-polynomials of lattice polytopes $P$ and $Q$ are LC-NIZ, then so is $h^*_{P\times Q}(t)$; the same holds for every finite Cartesian product [Cor. 5.1, LM26].

F. Liu surveys families of Ehrhart-positive polytopes, examples with negative Ehrhart coefficients, and open problems around Ehrhart positivity [Liu17]. M. Konoike studies how dilation affects magic positivity [Kon25]. Every polynomial with positive real coefficients becomes magic positive after sufficiently large positive dilation, but for each dimension $d\geq 3,$ some polytopes have fixed dilates that are not magic positive.

Example (Pitman–Stanley polytopes).

Let $y=(y_1,\dotsc,y_n)\in \setZ_{\gt{}0}^n.$ The Pitman–Stanley polytope is \[\Pi_n(y) \coloneqq \left\{ x\in \setR_{\geq 0}^n : x_1+\dotsb+x_k \leq y_1+\dotsb+y_k \text{ for } k=1,\dotsc,n \right\}.\] A broader marked-order model is obtained as follows. For $\mathbf y,\mathbf z\in\setZ_{\geq0}^k$ and $m\geq1,$ set $\widetilde y_i=y_1+\dotsb+y_i$ and $\widetilde z_i=z_1+\dotsb+z_i.$ The $m$-generalized Pitman–Stanley polytope $\operatorname{PS}_k^m(\mathbf y,\mathbf z)$ consists of the $(x_{i,j})\in\setR^{km}$ satisfying \[\widetilde z_i\leq x_{i,1}\leq\dotsb\leq x_{i,m} \leq\widetilde y_i, \qquad x_{i,j}\leq x_{i+1,j}.\] Here the first inequalities hold for $i\in[k],$ and the second for $i\in[k-1]$ and $j\in[m].$

Theorem (Jochemko–Menon [Thms. 3.2–3.3, JM26]).

For every $\mathbf y,\mathbf z\in\setZ_{\geq0}^k$ and $m\geq1,$ $\operatorname{PS}_k^m(\mathbf y,\mathbf z)$ is Ehrhart positive. When $\mathbf z=\mathbf0,$ the stronger multivariate statement holds: \[\left|\operatorname{PS}_k^m(\mathbf y,\mathbf0) \cap\setZ^{km}\right|\] is a polynomial in $\mathbf y$ with nonnegative rational coefficients.

N. Avila, L. Ferroni, and A. H. Morales prove that the Ehrhart polynomial $E_{\Pi_n(y)}(m)$ is magic positive [AFM26]. More precisely, if \[E_{\Pi_n(y)}(m) = \sum_{i=0}^n c_i m^i(1+m)^{n-i},\] then $n!c_i$ counts $y$-parking functions with exactly $i$ lucky cars in a modified parking protocol.

It follows from Brändén’s theorem that the $h^*$-polynomial $h^*_{\Pi_n(y)}(z)$ is real-rooted. In particular, its coefficients are log-concave and unimodal.

For computations, one may use the lattice-point function \[L_y(m) = \#\left\{ x\in\setZ_{\geq 0}^n : x_1+\dotsb+x_k \leq m(y_1+\dotsb+y_k) \text{ for } k=1,\dotsc,n \right\}.\] The corresponding $h^*$-coefficients are \[h^*_j = \sum_{i=0}^j (-1)^{j-i} \binom{n+1}{j-i} L_y(i).\] For example, \[\begin{array}{c|c} y & h^*_{\Pi_n(y)}(z) \\ \hline (1) & 1 \\ (2) & 1+z \\ (1,1) & 1+2z \\ (2,1) & 1+6z+z^2 \\ (1,1,1) & 1+10z+5z^2 \\ (2,1,1) & 1+24z+24z^2+z^3 . \end{array}\] See the page on parking functions for the classical parking function interpretation.

Example (Stasheff and symmetric edge polytopes).

M. Konoike studies two reflexive families [Kon24]. Let \[\operatorname{St}_d \coloneqq \operatorname{conv}\left( \{\pm e_i : 1\leq i \leq d\} \cup \{e_i+\dotsb+e_j : 1\leq i \lt j \leq d\} \right).\] Then $\operatorname{St}_d$ is the dual of the $d$-dimensional Stasheff polytope, and Konoike proves that the Ehrhart polynomial of the Stasheff polytope $\operatorname{St}_d^*$ is magic positive.

For the cycle graph $C_{d+1},$ the symmetric edge polytope is \[P_{C_{d+1}} \coloneqq \operatorname{conv}\left( \{\pm(e_i-e_{i+1}) : 1\leq i \leq d-1\} \cup \{\pm(e_d-e_1)\} \right).\] Writing the Ehrhart polynomial of the dual polytope as \[E_{P_{C_{d+1}}^*}(n) = \sum_{j=0}^d a_j n^j (n+1)^{d-j},\] the same paper proves that $a_i \gt{} 0$ and $a_{d-i} \gt{} 0$ for $i=0,1,2.$

#Further reading

Here we mainly discuss polytopes that arise in the area of algebraic combinatorics.

T. B. McAllister and H. O. Rochais study periods of Ehrhart coefficients of rational polytopes [MR18].
R. Davis, J. A. D. Loera, A. Garber, K. Jochemko, and J. Yu develop Ehrhart theory over abelian group rings [DLGJ+25].
M. v. Bell and C. Ceballos relate framing lattices and flow polytopes [BC25].
K. Rietsch and L. Williams study root polytopes, flow polytopes, and order polytopes [RW24].
M. Köbl studies roots of Ehrhart polynomials of symmetric edge polytopes from complete multipartite graphs [Ko24].

Bibliography

[Ale19]Per Alexandersson. Polytopes and large counterexamples. Experimental Mathematics, 28(1):115–120, 2019.

.bib

@article{Alexandersson2019CounterExamples,
author = {Per Alexandersson},
title = {Polytopes and Large Counterexamples},
journal = {Experimental Mathematics},
year  = {2019},
volume = {28},
number = {1},
pages = {115--120},
publisher = {Taylor \& Francis},
doi = {10.1080/10586458.2017.1361367},
URL2 = {http://www.tandfonline.com/doi/abs/10.1080/10586458.2017.1361367}
}

[Ale26]Per Alexandersson. Parking functions, Smirnov words, and noncrossing Chow polynomials. arXiv:2609.05131v1, 2026.

.bib

@article{Alexandersson2026ParkingSmirnov,
  author = {Per Alexandersson},
  title = {Parking functions, {S}mirnov words, and noncrossing {C}how
    polynomials},
  year = {2026},
  eprint = {2609.05131v1},
  url = {https://arxiv.org/abs/2609.05131v1},
  journal = {arXiv e-prints}
}

[AS23]Dave Anderson and Aniket Shah. Toric arc schemes and $q$-enumeration of lattice points. arXiv:2302.03761, 2023.

.bib

@article{AndersonShah2023x,
  author = {Dave Anderson and Aniket Shah},
  title = {Toric arc schemes and $q$-enumeration of lattice points},
  year = {2023},
  eprint = {2302.03761},
  url = {https://arxiv.org/abs/2302.03761},
  journal = {arXiv e-prints}
}

[Ard22]Federico Ardila. The bipermutahedron. Combinatorial Theory, 2(3), 2022. Article 1

.bib

@article{Ardila2022Bipermutahedron,
  author = {Federico Ardila},
  title = {The bipermutahedron},
  year = {2022},
  journal = {Combinatorial Theory},
  volume = {2},
  number = {3},
  note = {Article 1},
  doi = {10.5070/C62359149},
  eprint = {2008.02295v1}
}

[ABS11]Federico Ardila, Thomas Bliem and Dido Salazar. Gelfand–Tsetlin polytopes and Feigin–Fourier–Littelmann–Vinberg polytopes as marked poset polytopes. J. Comb. Theory, Ser. A, 118(8):2454–2462, 2011.

.bib

@article{ArdilaBliemSalazar2011,
  author = {Federico Ardila and Thomas Bliem and Dido Salazar},
  doi = {10.1016/j.jcta.2011.06.004},
  journal = {J. Comb. Theory, Ser. A},
  number = {8},
  pages = {2454--2462},
  title = {Gelfand--{T}setlin polytopes and {F}eigin--{F}ourier--{L}ittelmann--{V}inberg polytopes as marked poset polytopes},
  url2 = {http://dblp.uni-trier.de/db/journals/jct/jcta118.html\#ArdilaBS11},
  volume = {118},
  year = {2011}
}

[AFM26]Nicolas Avila, Luis Ferroni and Alejandro H. Morales. Luck and magic for Pitman–Stanley polytopes and parking functions. arXiv:2603.19194v2, 2026.

.bib

@article{AvilaFerroniMorales2026,
  author = {Nicolas Avila and Luis Ferroni and Alejandro H. Morales},
  title = {Luck and magic for {Pitman--Stanley} polytopes and parking functions},
  year = {2026},
  eprint = {2603.19194v2},
  archivePrefix = {arXiv},
  primaryClass = {math.CO}
}

[BDDL+24]Esme Bajo, Robert Davis, Jesús A. De Loera, Alexey Garber, Sofía Garzón Mora, Katharina Jochemko and Josephine Yu. Weighted Ehrhart theory: Extending Stanley’s nonnegativity theorem. Advances in Mathematics, 444:109627, 2024.

.bib

@article{BajoDavisLoeraGarberMoraJochemkoYu2024,
  author = {Bajo, Esme and Davis, Robert and De Loera, Jesús A. and Garber, Alexey and Garzón Mora, Sofía and Jochemko, Katharina and Yu, Josephine},
  title = {Weighted {E}hrhart theory: {E}xtending {S}tanley's nonnegativity theorem},
  year = {2024},
  journal = {Advances in Mathematics},
  volume = {444},
  pages = {109627},
  publisher = {Elsevier BV},
  doi = {10.1016/j.aim.2024.109627},
  url = {http://dx.doi.org/10.1016/j.aim.2024.109627},
  issn = {0001-8708},
  eprint = {2303.09614}
}

[BCGP25]Esther Banaian, Sunita Chepuri, Emily Gunawan and Jianping Pan. $c$-Birkhoff polytopes. arXiv:2504.07505, 2025.

.bib

@article{BanaianChepuriGunawanPan2025x,
  author = {Esther Banaian and Sunita Chepuri and Emily Gunawan and Jianping Pan},
  title = {$c$-{B}irkhoff polytopes},
  year = {2025},
  eprint = {2504.07505},
  url = {https://arxiv.org/abs/2504.07505},
  journal = {arXiv e-prints}
}

[BCGP26]Esther Banaian, Sunita Chepuri, Emily Gunawan and Jianping Pan. Type B c-Birkhoff polytopes are order polytopes. arXiv:2607.00646, 2026.

.bib

@article{BanaianChepuriGunawanPan2026x,
  author = {Esther Banaian and Sunita Chepuri and Emily Gunawan and Jianping Pan},
  title = {Type {B} c-{B}irkhoff polytopes are order polytopes},
  year = {2026},
  eprint = {2607.00646},
  url = {https://arxiv.org/abs/2607.00646},
  journal = {arXiv e-prints}
}

[BS07]Matthias Beck and Frank Sottile. Irrational proofs for three theorems of Stanley. European Journal of Combinatorics, 28(1):403–409, January 2007.

.bib

@article{Beck2007,
  title     = {Irrational proofs for three theorems of {S}tanley},
  volume    = {28},
  ISSN      = {0195-6698},
  url       = {http://dx.doi.org/10.1016/j.ejc.2005.06.003},
  DOI       = {10.1016/j.ejc.2005.06.003},
  number    = {1},
  journal   = {European Journal of Combinatorics},
  publisher = {Elsevier BV},
  author    = {Beck, Matthias and Sottile, Frank},
  year      = {2007},
  month     = jan,
  pages     = {403--409}
}

[BC25]Matias Bell and Cesar Ceballos. Framing Lattices and Flow Polytopes. arXiv:2512.20575, 2025.

.bib

@article{BellCeballos2025x,
  author = {Matias von Bell and Cesar Ceballos},
  title = {Framing {L}attices and {F}low {P}olytopes},
  year = {2025},
  eprint = {2512.20575},
  url = {https://arxiv.org/abs/2512.20575},
  journal = {arXiv e-prints}
}

[BLM26]Marie-Charlotte Brandenburg, Jesús A. De Loera and Chiara Meroni. From Ham-Sandwich to Centerpoints: Semialgebraic Algorithms for Cutting Polytopal Measures. arXiv:2607.02400, 2026.

.bib

@article{BrandenburgLoeraMeroni2026x,
  author = {Marie-Charlotte Brandenburg and Jes{\'u}s A. De Loera and
    Chiara Meroni},
  title = {From {H}am-{S}andwich to {C}enterpoints: {S}emialgebraic
    {A}lgorithms for {C}utting {P}olytopal {M}easures},
  year = {2026},
  eprint = {2607.02400},
  url = {https://arxiv.org/abs/2607.02400},
  journal = {arXiv e-prints}
}

[BJ26]Benjamin Braun and Aryaman Jal. Order polytopes of generalized snake posets are $h^*$-real-rooted. arXiv:2607.00922, 2026.

.bib

@article{BraunJal2026x,
  author = {Benjamin Braun and Aryaman Jal},
  title = {Order polytopes of generalized snake posets are {$h^*$}-real-rooted},
  year = {2026},
  eprint = {2607.00922},
  url = {https://arxiv.org/abs/2607.00922},
  journal = {arXiv e-prints}
}

[CS23]Jane Ivy Coons and Seth Sullivant. The $h^*$-polynomial of the order polytope of the zig-zag poset. The Electronic Journal of Combinatorics, 30(2):P2.44, 2023.

.bib

@article{CoonsSullivant2023,
  author = {Jane Ivy Coons and Seth Sullivant},
  title = {The {$h^*$}-polynomial of the order polytope of the zig-zag poset},
  journal = {The Electronic Journal of Combinatorics},
  volume = {30},
  number = {2},
  pages = {P2.44},
  year = {2023},
  doi = {10.37236/11526}
}

[DHY25]Rafael S. González D’León, Christopher R. H. Hanusa and Martha Yip. Permutation Flows I: Triangulations of Flow Polytopes (Research Announcement). arXiv:2512.04078, 2025.

.bib

@article{DLeonHanusaYip2025x,
  author = {Rafael S. Gonz{\'a}lez D'Le{\'o}n and Christopher R. H. Hanusa and
    Martha Yip},
  title = {Permutation {F}lows {I}: {T}riangulations of {F}low {P}olytopes
    ({R}esearch {A}nnouncement)},
  year = {2025},
  eprint = {2512.04078},
  url = {https://arxiv.org/abs/2512.04078},
  journal = {arXiv e-prints}
}

[DLGJ+25]Robert Davis, Jesús A. De Loera, Alexey Garber, Katharina Jochemko and Josephine Yu. Ehrhart Theory over Abelian Group Rings. arXiv:2511.10373, 2025.

.bib

@article{DavisLoeraGarberJochemkoYu2025x,
  author = {Robert Davis and Jes{\'u}s A. De Loera and Alexey Garber and
    Katharina Jochemko and Josephine Yu},
  title = {Ehrhart {T}heory over {A}belian {G}roup {R}ings},
  year = {2025},
  eprint = {2511.10373},
  url = {https://arxiv.org/abs/2511.10373},
  journal = {arXiv e-prints}
}

[EHR26]Richard Ehrenborg, Gábor Hetyei and Margaret Readdy. Parking trees and the toric $g$-vector of nestohedra. arXiv:2511.04815, 2026.

.bib

@article{EhrenborgHetyeiReaddy2026Parking,
  author = {Ehrenborg, Richard and Hetyei, G{\'a}bor and Readdy, Margaret},
  title = {Parking Trees and the Toric {$g$}-Vector of Nestohedra},
  journal = {arXiv e-prints},
  year = {2026},
  eprint = {2511.04815},
  url = {https://arxiv.org/abs/2511.04815}
}

[Fer26]Luis Ferroni. Unimodality shenanigans in Ehrhart theory. arXiv:2609.10513v1, 2026.

.bib

@article{Ferroni2026Unimodality,
  author = {Luis Ferroni},
  title = {Unimodality shenanigans in {E}hrhart theory},
  year = {2026},
  eprint = {2609.10513v1},
  url = {https://arxiv.org/abs/2609.10513v1},
  journal = {arXiv e-prints}
}

[FH24]Luis Ferroni and Akihiro Higashitani. Examples and counterexamples in ehrhart theory. EMS Surveys in Mathematical Sciences, 2024.

.bib

@article{FerroniHigashitani2024,
  author = {Luis Ferroni and Akihiro Higashitani},
  title = {Examples and counterexamples in Ehrhart theory},
  year = {2024},
  doi = {10.4171/emss/86},
  url = {https://doi.org/10.4171/emss/86},
  journal = {EMS Surveys in Mathematical Sciences}
}

[GK26]Peter L. Guo and Mingyang Kang. Palindromic real-rooted polynomials and $h$-vectors of polytopes. arXiv:2609.00781v1, 2026.

.bib

@article{GuoKang2026x,
  author = {Peter L. Guo and Mingyang Kang},
  title = {Palindromic real-rooted polynomials and $h$-vectors of polytopes},
  year = {2026},
  eprint = {2609.00781v1},
  url = {https://arxiv.org/abs/2609.00781v1},
  journal = {arXiv e-prints}
}

[HM08]Christian Haase and Tyrrell B. McAllister. Quasi-period collapse and ${GL}_n({Z})$-scissors congruence in rational polytopes. Contemporary mathematic, 452:115–122, 2008.

.bib

@inproceedings{HaaseMcAllister2007,
  booktitle = {Contemporary Mathematic},
  doi = {10.1090/conm/452/08777},
  url2 = {https://doi.org/10.1090/conm/452/08777},
  year = {2008},
  volume= {452},
  publisher = {American Mathematical Society},
  pages = {115--122},
  author = {Christian Haase and Tyrrell B. McAllister},
  title = {Quasi-period collapse and ${GL}_n({Z})$-scissors congruence in rational polytopes}
}

[HLV23]Mitsuki Hanada, John Lentfer and Andrés R. Vindas-Meléndez. Generalized Parking Function Polytopes. Annals of Combinatorics, 28(2):575–613, 2023.

.bib

@article{HanadaLentferVindasMelendez2023,
  author = {Hanada, Mitsuki and Lentfer, John and Vindas-Meléndez, Andrés R.},
  title = {Generalized {P}arking {F}unction {P}olytopes},
  year = {2023},
  journal = {Annals of Combinatorics},
  volume = {28},
  number = {2},
  pages = {575--613},
  publisher = {Springer},
  doi = {10.1007/s00026-023-00671-1},
  url = {https://doi.org/10.1007/s00026-023-00671-1},
  eprint = {2212.06885}
}

[Hib92]Takayuki Hibi. Algebraic combinatorics on convex polytopes. Carslaw Publications, 1992.

.bib

@book{Hibi92,
	author = {Takayuki Hibi},
	title = {Algebraic Combinatorics on Convex Polytopes},
	publisher = {Carslaw Publications},
	year = {1992}
}

[HJM19]Akihiro Higashitani, Katharina Jochemko and Mateusz Michałek. Arithmetic aspects of symmetric edge polytopes. Mathematika, 65(3):763–784, 2019.

.bib

@article{HigashitaniJochemkoMichalek2019,
  author = {Akihiro Higashitani and Katharina Jochemko and Mateusz
    Micha{\l}ek},
  title = {Arithmetic aspects of symmetric edge polytopes},
  year = {2019},
  journal = {Mathematika},
  volume = {65},
  number = {3},
  pages = {763--784},
  doi = {10.1112/S0025579319000147},
  eprint = {1807.07678v1}
}

[JM26]Katharina Jochemko and Krishna Menon. Ehrhart positivity for marked order polytopes. arXiv:2604.08394v2, 2026.

.bib

@article{JochemkoMenon2026x,
  author = {Katharina Jochemko and Krishna Menon},
  title = {Ehrhart positivity for marked order polytopes},
  year = {2026},
  eprint = {2604.08394v2},
  url = {https://arxiv.org/abs/2604.08394v2},
  journal = {arXiv e-prints}
}

[Ko24]Max Kölbl. On a Conjecture Concerning the Roots of Ehrhart Polynomials of Symmetric Edge Polytopes from Complete Multipartite Graphs. arXiv:2404.02136, 2024.

.bib

@article{Kolbl2024x,
  author = {Max K{\"o}lbl},
  title = {On a {C}onjecture {C}oncerning the {R}oots of {E}hrhart
    {P}olynomials of {S}ymmetric {E}dge {P}olytopes from {C}omplete
    {M}ultipartite {G}raphs},
  year = {2024},
  eprint = {2404.02136},
  url = {https://arxiv.org/abs/2404.02136},
  journal = {arXiv e-prints}
}

[Kon24]Masato Konoike. A new class of magic positive ehrhart polynomials of reflexive polytopes. arXiv:2409.16648, 2024.

.bib

@article{Konoike2024x,
  author = {Masato Konoike},
  title = {A new class of magic positive Ehrhart polynomials of reflexive polytopes},
  year = {2024},
  eprint = {2409.16648},
  url = {https://arxiv.org/abs/2409.16648},
  journal = {arXiv e-prints}
}

[Kon25]Masato Konoike. On the magic positivity of Ehrhart polynomials of dilated polytopes. arXiv:2504.21395, 2025.

.bib

@article{Konoike2025x,
  author = {Masato Konoike},
  title = {On the magic positivity of {E}hrhart polynomials of dilated polytopes},
  year = {2025},
  eprint = {2504.21395},
  url = {https://arxiv.org/abs/2504.21395},
  journal = {arXiv e-prints}
}

[Li12]Nan Li. Ehrhart $h^*$-vectors of hypersimplices. Discrete & Computational Geometry, 48(4):847–878, 2012.

.bib

@article{Li2012,
  author = {Nan Li},
  title = {Ehrhart {$h^*$}-vectors of hypersimplices},
  year = {2012},
  journal = {Discrete \& Computational Geometry},
  volume = {48},
  number = {4},
  pages = {847--878},
  doi = {10.1007/s00454-012-9452-2},
  url = {https://doi.org/10.1007/s00454-012-9452-2}
}

[Liu17]Fu Liu. On positivity of Ehrhart polynomials. arXiv:1711.09962, 2017.

.bib

@article{Liu2017,
Author = {Fu Liu},
Title = {On positivity of {E}hrhart polynomials},
Year = {2017},
Eprint = {1711.09962},
  url = {https://arxiv.org/abs/1711.09962},
journal = {arXiv e-prints}
}

[LM26]Yanxin Liu and Jianxi Mao. Preservation of log-concavity under Hadamard products. arXiv:2609.11589v1, 2026.

.bib

@article{LiuMao2026Hadamard,
  author = {Yanxin Liu and Jianxi Mao},
  title = {Preservation of log-concavity under {H}adamard products},
  year = {2026},
  eprint = {2609.11589v1},
  url = {https://arxiv.org/abs/2609.11589v1},
  journal = {arXiv e-prints}
}

[LMD19]Ricky I. Liu, Karola Mészáros and Avery St. Dizier. Gelfand–Tsetlin Polytopes: A Story of Flow and Order Polytopes. SIAM Journal on Discrete Mathematics, 33(4):2394–2415, 2019.

.bib

@article{LiuMeszarosDizier2019flow,
  author = {Liu, Ricky I. and Mészáros, Karola and Dizier, Avery St.},
  title = {Gelfand--{T}setlin {P}olytopes: {A} {S}tory of {F}low and {O}rder
    {P}olytopes},
  year = {2019},
  journal = {SIAM Journal on Discrete Mathematics},
  volume = {33},
  number = {4},
  pages = {2394--2415},
  publisher = {Society for Industrial \& Applied Mathematics},
  doi = {10.1137/19m1251242},
  url = {https://doi.org/10.1137/19m1251242},
  eprint = {1903.08275}
}

[LT25]Fu Liu and Warut Thawinrak. Parking Function Polytopes. arXiv:2512.14199, 2025.

.bib

@article{LiuThawinrak2025x,
  author = {Fu Liu and Warut Thawinrak},
  title = {Parking {F}unction {P}olytopes},
  year = {2025},
  eprint = {2512.14199},
  url = {https://arxiv.org/abs/2512.14199},
  journal = {arXiv e-prints}
}

[LT19]Fu Liu and Akiyoshi Tsuchiya. Stanley’s non-Ehrhart-positive order polytopes. Advances in Applied Mathematics, 108:1–10, 2019.

.bib

@article{LiuTsuchiya2018,
Author = {Fu Liu and Akiyoshi Tsuchiya},
Title = {Stanley's non-{E}hrhart-positive order polytopes},
Year = {2019},
Eprint = {1806.08403},
url = {https://arxiv.org/abs/1806.08403},
journal = {Advances in Applied Mathematics},
volume = {108},
pages = {1--10},
doi = {10.1016/j.aam.2019.03.004}
}

[LXZ26]Feihu Liu, Guoce Xin and Zihao Zhang. Order polytopes of dimension $\leq 13$ are Ehrhart positive. Discrete Mathematics, 349:114833, 2026.

.bib

@article{LiuXinZhang2026Dimension13,
  author = {Feihu Liu and Guoce Xin and Zihao Zhang},
  title = {Order Polytopes of Dimension $\leq 13$ are {E}hrhart Positive},
  journal = {Discrete Mathematics},
  volume = {349},
  pages = {114833},
  year = {2026},
  doi = {10.1016/j.disc.2025.114833},
  eprint = {2412.07164},
  archivePrefix = {arXiv}
}

[LEKW24]Jesús A. De Loera, Laura Escobar, Nathan Kaplan and Chengyang Wang. Sums of Weighted Lattice Points of Polytopes. arXiv:2402.11328, 2024.

.bib

@article{LoeraEscobarKaplanWang2024x,
  author = {Jesús A. De Loera and Laura Escobar and Nathan Kaplan and Chengyang Wang},
  title = {Sums of {W}eighted {L}attice {P}oints of {P}olytopes},
  year = {2024},
  eprint = {2402.11328},
  url = {https://arxiv.org/abs/2402.11328},
  journal = {arXiv e-prints}
}

[LSML26]Teemu Lundström and Leonardo Saud Maia Leite. Order polytopes of crown posets. European Journal of Combinatorics, 133:104304, 2026.

.bib

@article{LundstromLeite2026,
  author = {Lundstr{\"o}m, Teemu and Saud Maia Leite, Leonardo},
  title = {Order polytopes of crown posets},
  year = {2026},
  journal = {European Journal of Combinatorics},
  volume = {133},
  pages = {104304},
  publisher = {Elsevier BV},
  doi = {10.1016/j.ejc.2025.104304},
  url = {http://dx.doi.org/10.1016/j.ejc.2025.104304},
  issn = {0195-6698},
  eprint = {2504.05123}
}

[Mac71]I. G. Macdonald. Polynomials associated with finite cell-complexes. Journal of the London Mathematical Society, s2-4(1):181–192, July 1971.

.bib

@article{Macdonald1971,
  title     = {Polynomials Associated with Finite Cell-Complexes},
  volume    = {s2-4},
  ISSN      = {0024-6107},
  url       = {http://dx.doi.org/10.1112/jlms/s2-4.1.181},
  DOI       = {10.1112/jlms/s2-4.1.181},
  number    = {1},
  journal   = {Journal of the London Mathematical Society},
  publisher = {Wiley},
  author    = {Macdonald, I. G.},
  year      = {1971},
  month     = jul,
  pages     = {181--192}
}

[MR18]Tyrrell B. McAllister and Hélène O. Rochais. Periods of Ehrhart Coefficients of Rational Polytopes. The Electronic Journal of Combinatorics, 25(1), 2018.

.bib

@article{McAllisterRochais2018,
  author = {Tyrrell B. McAllister and H{\'e}l{\`e}ne O. Rochais},
  title = {Periods of {E}hrhart {C}oefficients of {R}ational {P}olytopes},
  year = {2018},
  journal = {The Electronic Journal of Combinatorics},
  volume = {25},
  number = {1},
  publisher = {The Electronic Journal of Combinatorics},
  doi = {10.37236/6059},
  url = {https://doi.org/10.37236/6059}
}

[MV25]Krishna Menon and Emil Verkama. Interpreting the Ehrhart coefficients of cross-polytopes. arXiv:2512.08669, 2025.

.bib

@article{MenonVerkama2025x,
  author = {Krishna Menon and Emil Verkama},
  title = {Interpreting the {E}hrhart coefficients of cross-polytopes},
  year = {2025},
  eprint = {2512.08669},
  url = {https://arxiv.org/abs/2512.08669},
  journal = {arXiv e-prints}
}

[OT21]Hidefumi Ohsugi and Akiyoshi Tsuchiya. Symmetric edge polytopes and matching generating polynomials. Combinatorial Theory, 1, 2021. Article 9

.bib

@article{OhsugiTsuchiya2021,
  author = {Hidefumi Ohsugi and Akiyoshi Tsuchiya},
  title = {Symmetric edge polytopes and matching generating polynomials},
  year = {2021},
  journal = {Combinatorial Theory},
  volume = {1},
  note = {Article 9},
  doi = {10.5070/C61055371},
  eprint = {2008.08621v2}
}

[Pak00]Igor Pak. Four questions on Birkhoff polytope. Annals of Combinatorics, 4(1):83–90, 2000.

.bib

@article{Pak2000,
	year={2000},
	issn={0218-0006},
	journal={Annals of Combinatorics},
	volume={4},
	number={1},
	doi={10.1007/PL00001277},
	title={Four Questions on {B}irkhoff Polytope},
	publisher={Birkh{\"{a}}user Verlag},
	keywords={Keywords: Birkhoff polytope, simplex method, random walk, symmetric group, mixing time},
	author={Igor Pak},
	pages={83--90}
}

[RR24]Victor Reiner and Brendon Rhoades. Harmonics and graded Ehrhart theory. arXiv:2407.06511, 2024.

.bib

@article{ReinerRhoades2024x,
  author = {Victor Reiner and Brendon Rhoades},
  title = {Harmonics and graded {E}hrhart theory},
  year = {2024},
  eprint = {2407.06511},
  url = {https://arxiv.org/abs/2407.06511},
  journal = {arXiv e-prints}
}

[RW24]Konstanze Rietsch and Lauren Williams. Root polytopes, flow polytopes, and order polytopes. arXiv:2406.15803, 2024.

.bib

@article{RietschWilliams2024x,
  author = {Konstanze Rietsch and Lauren Williams},
  title = {Root polytopes, flow polytopes, and order polytopes},
  year = {2024},
  eprint = {2406.15803},
  url = {https://arxiv.org/abs/2406.15803},
  journal = {arXiv e-prints}
}

[Sta93]Richard P. Stanley. A monotonicity property of h-vectors and h*-vectors. European Journal of Combinatorics, 14(3):251–258, May 1993.

.bib

@article{Stanley1993,
  title   = {A Monotonicity Property of h-vectors and h*-vectors},
  volume  = {14},
  ISSN    = {0195-6698},
  url     = {http://dx.doi.org/10.1006/eujc.1993.1028},
  DOI     = {10.1006/eujc.1993.1028},
  number  = {3},
  journal = {European Journal of Combinatorics},
  publisher = {Elsevier BV},
  author  = {Stanley, Richard P.},
  year    = {1993},
  month   = may,
  pages   = {251--258}
}

[Sta86]Richard P. Stanley. Two poset polytopes. Discrete & Computational Geometry, 1(1):9–23, 1986.

.bib

@Article{Stanley86TwoPosetPolytopes,
author = {Richard P. Stanley},
title={Two poset polytopes},
journal={Discrete \& Computational Geometry},
year={1986},
volume={1},
number={1},
pages={9--23},
issn={1432-0444},
doi={10.1007/BF02187680}
}

[Sta11]Richard P. Stanley. Enumerative Combinatorics: Volume 1. Cambridge University Press, Second, 2011.

.bib

@book{StanleyEC1,
 author = {Richard P. Stanley},
 title = {Enumerative {C}ombinatorics: {V}olume 1},
 year = {2011},
 isbn = {9781107602625},
 edition = {Second},
 doi = {10.1017/CBO9781139058520},
 publisher = {Cambridge University Press}
}

[Str24]Jan Stricker. 2-levelness of Marked Poset Polytopes and the Ehrhart polynomial. arXiv:2401.07492, 2024.

.bib

@article{Stricker2024x,
  author = {Jan Stricker},
  title = {2-levelness of {M}arked {P}oset {P}olytopes and the {E}hrhart polynomial},
  year = {2024},
  eprint = {2401.07492},
  url = {https://arxiv.org/abs/2401.07492},
  journal = {arXiv e-prints}
}

[Xia26]Qiqi Xiao. The real-rootedness of the toric $g$-contribution polynomials. arXiv:2609.01086v1, 2026.

.bib

@article{Xiao2026x,
  author = {Qiqi Xiao},
  title = {The real-rootedness of the toric $g$-contribution polynomials},
  year = {2026},
  eprint = {2609.01086v1},
  url = {https://arxiv.org/abs/2609.01086v1},
  journal = {arXiv e-prints}
}

[Zie95]Günter M. Ziegler. Lectures on polytopes. Springer-Verlag, 1995.

.bib

@book{ZieglerLecturesOnPolytopes95,
  added-at = {2009-12-02T16:16:33.000+0100},
  address = {New York},
  author = {G{\"{u}}nter M. Ziegler},
  booktitle = {Graduate texts in mathematics, 152},
  issn = {0387943293  9780387943299  3540943293  9783540943297  038794365X  9780387943657  354094365X  9783540943655},
  keywords = {imported nearest-neighbor polytopes},
  publisher = {Springer-Verlag},
  title = {Lectures on polytopes},
  url = {http://www.worldcat.org/search?qt=worldcat_org_all&q=9780387943657},
  year = {1995}
}

The symmetric functions catalog

#Polytopes

#Face \(h\)- and toric \(g\)-polynomials

#The Ehrhart polynomial and \(h^*\)-vector

#Ehrhart–Macdonald reciprocity

#Stanley’s monotonicity theorem

#Families of polytopes

#Hypersimplices

#Order polytopes

#Chain polytopes

#Marked order polytopes

#Cross-polytopes

#Root polytopes

#Flow polytopes

#The permutohedron

#Generalized permutohedra

#Birkhoff polytopes

#Edge polytopes

#Perfect matching polytopes

#Associahedra

#Parking function polytopes

#The integer decomposition property

#Reflexive and Gorenstein polytopes

#Magic positivity and reflexive polytopes

#Further reading

Bibliography