Enriched order polytopes and Enriched Hibi rings

Stanley introduced two classes of lattice polytopes associated to posets, which are called the order polytope ${\mathcal O}_P$ and the chain polytope ${\mathcal C}_P$ of a poset $P$. It is known that, given a poset $P$, the Ehrhart polynomials of ${\mathcal O}_P$ and ${\mathcal C}_P$ are equal to the order polynomial of $P$ that counts the $P$-partitions. In this paper, we introduce the enriched order polytope of a poset $P$ and show that it is a reflexive polytope whose Ehrhart polynomial is equal to that of the enriched chain polytope of $P$ and the left enriched order polynomial of $P$ that counts the left enriched $P$-partitions, by using the theory of Gr\"{o}bner bases. The toric rings of enriched order polytopes are called enriched Hibi rings. It turns out that enriched Hibi rings are normal, Gorenstein, and Koszul. The above result implies the existence of a bijection between the lattice points in the dilations of ${\mathcal O}^{(e)}_P$ and ${\mathcal C}^{(e)}_P$. Towards such a bijection, we give the facet representations of enriched order and chain polytopes.


INTRODUCTION
A lattice polytope in R n is a convex polytope all of whose vertices are in Z n . In [18], Stanley introduced a class of lattice polytopes associated to finite partially ordered sets (poset for short). Let (P, < P ) be a finite poset on [n] := {1, . . . , n}. The order polytope O P of P is the convex polytope consisting of the set of points (x 1 , . . . , x n ) ∈ R n such that (1) 0 ≤ x i ≤ 1 for 1 ≤ i ≤ n, (2) x i ≤ x j if i < P j.
Then O P is a lattice polytope of dimension n. In fact, each vertex of O P corresponds to a filter of P. Here, a subset F of P is called a filter of P if i ∈ F and j ∈ P together with i < P j guarantee j ∈ F. For a subset X ⊂ [n], we define the (0, 1)-vector e X := ∑ i∈X e i , where e 1 , . . . , e n are the canonical unit coordinate vectors of R n . Then one has O P ∩ Z n = {e F : F ∈ F (P)}, where F (P) is the set of filters of P. Moreover, there is a close interplay between the combinatorial structure of P and the geometric structure of O P . Assume that P is naturally labeled, i.e., i < j if i < P j. Let Z ≥0 be the set of nonnegative integers. A map f : P → Z ≥0 is called a P-partition if for all x, y ∈ P with x < P y, f satisfies f (x) ≤ f (y). We identify a P-partition f with a lattice point ( f (1), . . ., f (n)) ∈ Z n . Since every P-partition f : P → Z ≥0 with f (i) ≤ 1 is a filter of P, the set of P-partitions f : P → Z ≥0 with f (i) ≤ 1 coincides with O P ∩ Z n . Moreover, the set of P-partitions f : P → Z ≥0 with f (i) ≤ m coincides with mO P ∩ Z n for 0 < m ∈ Z.
Here, for a convex polytope P ⊂ R n , mP := {mx : x ∈ P} is the m-th dilated polytope.
In the present paper, we define a new class of lattice polytopes associated to finite posets from a viewpoint of the theory of enriched P-partitions. For a filter F of P, we set F min := min(F) and F comin := F \ F min , where min(F) is the set of minimal elements of F. For a subset X = {i 1 , . . ., i r } ⊂ [n] and a vector ε = (ε 1 , . . . , ε r ) ∈ {−1, 1} r , we define the (−1, 0, 1)-vector e ε X := ∑ r j=1 ε j e i j . The enriched order polytope O (e) P ⊂ R n of a finite (not necessarily naturally labeled) poset P on [n] is the lattice polytope of dimension n which is the convex hull of (1) {e ε F min + e F comin : F ∈ F (P), ε ∈ {−1, 1} |F min | }.
Then O (e) P ∩ Z n coincides with the set (1) above (Lemma 4.1). Now, we discuss a relation between O (e) P and the theory of enriched P-partitions. Again, we assume that P is naturally labeled. A map f : P → Z \ {0} is called an enriched P-partition ( [19]) if, for all x, y ∈ P with x < P y, f satisfies On the other hand, Petersen [17] introduced slightly different notion "left enriched Ppartitions" as follows. A map f : P → Z is called a left enriched P-partition if, for all x, y ∈ P with x < P y, f satisfies the following conditions: Then the set of left enriched P-partitions f : P → Z with | f (i)| ≤ 1 coincides with O (e) P ∩ Z n . Contrary to the case of order polytopes, the set of left enriched P-partitions f : P → Z with | f (i)| ≤ m does not always coincide with the set of lattice points mO (e) P ∩ Z n for m > 1 (Example 4.2). However, we will show that the number Ω where P is the dual poset of P.
In this paper, in order to show Theorem 1.1, we investigate the toric ring of the enriched order polytope O (e) P . In [5], Hibi studied the toric ring of the order polytope O P . The toric ideal I O P possesses a squarefree quadratic Gröbner basis, that is a Gröbner basis consisting of binomials whose initial monomials are squarefree and of degree 2. This implies that the toric ring K First, in Section 2, we introduce known results on two poset polytopes introduced by Stanley [18], that is, the order polytope O P and the chain polytope C P of a poset P. A squarefree quadratic Gröbner basis of the toric ideal of each of O P , C P and its applications will be extended to "enriched case" in the following sections. In Section 3, we study the notion of enriched chain polytopes C (e) P ( [16]) because we need to compare the toric ideals of enriched order polytopes and that of enriched chain polytopes in order to prove Theorem 1.1. In Section 4, we discuss fundamental properties of enriched order polytopes. In Section 5, we study the toric ideals of enriched order polytopes and their applications. By proving that the toric ideal of O (e) P possesses a squarefree quadratic Gröbner basis consisting of binomials whose initial monomials do not contain the variable corresponding to the origin, we show Theorem 1.2 (Corollary 5.3). Moreover, by comparing the initial ideals of toric ideals of two enriched poset polytopes (Theorem 5.4), we will complete the proof of Theorem 1.1. Note that Theorem 1.1 implies the existence of a bijection between mO (e) P ∩ Z n and mC (e) P ∩ Z n . In Section 6, towards such a bijection, we consider an elementary geometric property, the facet representations of enriched order and chain polytopes (Proposition 6.1, Theorem 6.2). The number of facets is discussed in Corollary 6.3, and Proposition 6.5. Finally, we show that O (e) P is rarely unimodularly equivalent to C (e) P (Proposition 6.6). Acknowledgment. The authors are grateful to an anonymous referee for his useful comments. In particular, the last section was added following his advice. The authors were partially supported by JSPS KAKENHI 18H01134 and 16J01549.

TWO POSET POLYTOPES
In this section, we review properties of order polytopes and chain polytopes. Let (P, < P ) be a finite poset on [n]. Recall that the order polytope O P ⊂ R n is the convex hull of In [18], Stanley introduced another lattice polytope associated to P as well as the order polytope O P . An antichain of P is a subset of P consisting of pairwise incomparable elements of P. Let A (P) denote the set of antichains of P. Note that the empty set / 0 is an antichain of P. The chain polytope C P ⊂ R n of P is the convex hull of Then C P is a lattice polytope of dimension n. The order polytope O P and the chain polytope C P have similar properties.
First, we study the Ehrhart polynomials of O P and C P . Let P ⊂ R n be a general lattice polytope of dimension n. Given a positive integer m, we define L P (m) = |mP ∩ Z n |. 3 The study on L P (m) originated in Ehrhart [3] who proved that L P (m) is a polynomial in m of degree n with the constant term 1. Moreover, the leading coefficient of L P (m) coincides with the usual Euclidean volume of P. We say that L P (m) is the Ehrhart polynomial of P. An Ehrhart polynomial often coincides with a counting function of a combinatorial object. A map f : P → Z ≥0 is called an order preserving map if for all x, y ∈ P with x < P y, f satisfies f (x) ≤ f (y). Let Ω P (m) denote the number of order preserving maps f : P → Z ≥0 with f (i) ≤ m. Then Ω P (m) is a polynomial in m of degree n and called the order polynomial of P. Stanley showed a relation between the Ehrhart polynomials of O P and C P and the order polynomial Ω P (m). In fact, On the other hand, O P and C P are not always unimodularly equivalent. Here, two lattice polytopes P, Q ⊂ R n of dimension n are unimodularly equivalent if there exist a unimodular matrix U ∈ Z n×n and a lattice point w ∈ Z n such that Q = f U (P) + w, where f U is the linear transformation in R n defined by U , i.e., f U (x) = xU for all x ∈ R n . In [7], Hibi and Li characterized when O P and C P are unimodularly equivalent. In fact, (i) The order polytope O P and the chain polytope C P are unimodularly equivalent; (ii) The number of the facets of O P is equal to that of C P ; (iii) The following poset is not a subposet of P.
Next, we review the toric ideals of order polytopes and chain polytopes. First, we recall basic materials and notation on toric ideals. Let K[t ±1 , s] = K[t ±1 1 , . . . ,t ±1 n , s] be the Laurent polynomial ring in n + 1 variables over a field K. If a = (a 1 , . . ., a n ) ∈ Z n , then t a s is the Laurent monomial t a 1 1 · · ·t a n n s ∈ K[t ±1 , s]. Let P ⊂ R n be a lattice polytope and P ∩ Z n = {a 1 , . . . , a d }. Then, the toric ring of P is the subalgebra K[P] of K[t ±1 , s] generated by {t a 1 s, . . ., t a d s} over K. We regard K[P] as a homogeneous algebra by setting each deg t a i s = 1. Let K[x] = K[x 1 , . . ., x d ] denote the polynomial ring in d variables over K with each deg(x i ) = 1. The toric ideal I P of P is the kernel of the surjective homomorphism π : K[x] → K[P] defined by π(x i ) = t a i s for 1 ≤ i ≤ d. It is known that I P is generated by homogeneous binomials. See, e.g., [4,20]. Now, we study the toric ideals of O P and C P . Remark that O P and O P are unimodularly equivalent and C P = C P . A subset I of P is called a poset ideal of P if i ∈ I and j ∈ P together with i > P j guarantee j ∈ I. Let J (P) denote the set of poset ideals of P, ordered by inclusion. If I ∈ J (P) and J ∈ J (P) are incomparable in J (P), then we write I ≁ J. Then the order polytope O P is the convex hull of {e I : I ∈ J (P)}. 4 Let R[O] denote the polynomial ring over K in variables x I , where I ∈ J (P). In particular, the origin corresponds to the variable x / 0 . Then the toric ideal I O P is the kernel of the ring homomorphism π O : In [5], Hibi essentially proved that I O P possesses a squarefree quadratic Gröbner basis. In fact, . Work with the same notation as above. Then For a poset ideal I of P, we denote max(I) the set of maximal elements of I. Then max(I) is an antichain of P and every antichain of P is the set of maximal elements of a poset ideal. On the other hand, for an antichain A of P, the poset ideal of P generated by A is the smallest poset ideal of P which contains A. Every poset ideal of P can be obtained by this way. Hence J (P) and A (P) have a one-to-one correspondence. Let R[C ] denote the polynomial ring over K in variables x max(I) , where I ∈ J (P). Then the toric ideal I C P is the kernel of the ring homomorphism π C : Given poset ideals I, J ∈ J (P), let I * J denote the poset ideal of P generated by max(I ∩ J) ∩ (max(I) ∪ max(J)). Note that I * J ⊂ I ∩ J. The following lemma is fundamental and important.
Lemma 2.4. Let P be a finite poset and I, J ∈ J (P). For p ∈ max(I) \ max(J), the following conditions are equivalent: Then there exists an element q ∈ I ∪ J such that p < P q. If q belongs to I, then p / ∈ max(I), a contradiction. Thus q ∈ J, and hence p ∈ I ∩ J. If p / ∈ max(I ∩ J) holds, then there exists an element q ′ ∈ I ∩ J such that p < P q ′ . This contradicts to the hypothesis p ∈ max(I). Thus (iv) ⇒ (ii) holds. Finally, we have (ii) ⇔ (iii) by max(I * J) = max(I ∩ J) ∩ (max(I) ∪ max(J)).
In [6], Hibi and Li essentially proved that I C P possesses a squarefree quadratic Gröbner basis. In fact, Proposition 2.5 ( [6]). Work with the same notation as above. Then is a Gröbner basis of I C P with respect to a reverse lexicographic order < C . Moreover, R[C ]/I C P is a normal Cohen-Macaulay domain and Koszul.
From Propositions 2.3 and 2.5 we can prove the following. Proposition 2.6. Work with the same notation as above. Then one has .
Proof. From Propositions 2.3 and 2.5, we have Hence it follows that the map . Therefore, the first claim follows. Since both in < O (I O P ) and in < C (I C P ) are squarefree, both O P and C P possesses a unimodular triangulation, and hence the Ehrhart polynomial coincides with the Hilbert polynomial of its toric ring for each of O P and C P (see [4,Section 4.2] or [20,Chapters 8 and 13]). Moreover, for an ideal I of K[x] and a monomial order < on K[x], the Hilbert polynomial of K[x]/I is equal to that of K[x]/in < (I). Therefore, the second claim follows.

ENRICHED CHAIN POLYTOPES
In this section, we recall the definition and properties of enriched chain polytopes given in [16]. Let (P, < P ) be a finite poset on [n]. The enriched chain polytope C (e) P is a lattice polytope of dimension n. It is easy to see that C A lattice polytope P ⊂ R n of dimension n is called reflexive if the origin of R n is a unique lattice point belonging to the interior of P and its dual polytope is also a lattice polytope, where x, y is the usual inner product of R n . It is known that reflexive polytopes correspond to Gorenstein toric Fano varieties, and they are related to mirror symmetry (see, e.g., [1,2]). In each dimension there exist only finitely many reflexive polytopes up to unimodular equivalence ( [13]) and all of them are known up to dimension 4 ( [12]). Recently, several classes of reflexive polytopes were constructed by an algebraic technique on Gröbner bases (c.f., [10,11,15]). The algebraic technique is based on the following lemma that follows from the argument in [9, Proof of Lemma 1.1].
Lemma 3.1. Let P ⊂ R n be a lattice polytope of dimension n such that the origin of R n is contained in its interior. Suppose that any lattice point in Z n is a linear integer combination of the lattice points in P. If there exists a monomial order such that the initial ideal of I P is generated by squarefree monomials which do not contain the variable corresponding to the origin, then P is reflexive and has a regular unimodular triangulation. Moreover, K[P] is a normal Gorenstein domain.
In order to use Lemma 3.1 for enriched chain polytopes C (e) P , we study the toric ideal of C (e) Then the toric ideal I . It is known [20, Proposition 1.11] that there exists a nonnegative weight vector w C ∈ R |J (P)| such that in w C (I C P ) = in < C (I C P ). Then we define the weight vector w C (e) on R[C (e) ] such that the weight of each variable x ε A with respect to w C (e) is the weight of the variable x ε + A with respect to w C . In addition, let w ♯ be the weight vector on R[C (e) ] such that the weight of each variable x ε A with respect to w ♯ is |A|. Fix any monomial order ≺ on K[C (e) ] as a tie-breaker. Let < C (e) be a monomial order on R[C (e) ] such that u < C (e) v if and only if one of the following holds: • The weight of u is less than that of v with respect to w ♯ ; • The weight of u is the same as that of v with respect to w ♯ , and the weight of u is less than that of v with respect to w C (e) ; • The weight of u is the same as that of v with respect to w ♯ and w C (e) , and u ≺ v. (c) For any p ∈ max(J) \ max(I), we have ε ′ p = µ p if p ∈ max(I ∪ J), µ ′ p if p ∈ max(I * J). Then G C (e) is a Gröbner basis of I C (e) P with respect to a monomial order < C (e) . The initial monomial of each binomial is the first monomial. In particular, the initial ideal is generated by squarefree quadratic monomials which do not contain the variable x 0  Next, we study Ehrhart polynomials of enriched chain polytopes. Assume that P is naturally labeled. Let Ω

FUNDAMENTAL PROPERTIES OF ENRICHED ORDER POLYTOPES
In this section, we discuss some fundamental properties of enriched order polytopes. First, we consider the set of lattice points in enriched order polytopes.
In addition, the origin is the unique interior lattice point in O Then each a i is a (−1, 0, 1)-vector, and hence so is x. It is easy to see that x k = 1 (resp. x k = −1) if and only if k-th component of a i is equal to 1 (resp. −1) for all i = 1, 2, . . ., s. Suppose that k < P ℓ. If x k = 0, then |x k | ≤ |x ℓ | and the equality holds if and only if x ℓ = 0. Suppose that |x k | = 1. Then k-th component of a i is equal to x k for all i = 1, 2, . . ., s. Since each a i is a left enriched P-partition, ℓ-th component of a i is equal to 1 for all i = 1, 2, . . ., s. Hence x ℓ = 1. In particular, |x k | = |x ℓ | and x ℓ ≥ 0. Thus x is a left enriched P-partition, that is, x belongs to X .
Since O P which contains 0. Let H = {y ∈ R n : a, y = 0} with 0 = a = (a 1 , . . . , a n ) ∈ R n be the supporting hyperplane of F and let P ′ = {i ∈ P : a i = 0} ( = / 0) be a subposet of P. We may assume that i ∈ max(P ′ ) satisfies a i > 0. Let F = { j ∈ P : i ≤ P j} be a filter of P. Then F min = {i} and hence y = e (1) F min +e F comin satisfies a, y = a i > 0 and y ′ = e (−1) F min +e F comin satisfies a, y ′ = −a i < 0. This contradicts that H is a supporting hyperplane of O (e) P .
Next, we consider lattice points in the dilated polytopes of an enriched order polytope. The following example shows that, contrary to the case of order polytopes, the set of left enriched P-partitions f : P → Z wtih | f (i)| ≤ m does not always coincide with the set of lattice points mO where I, J ∈ J (P), ε p = µ p , and p ∈ max(I) ∩ max(J), together with all binomials . On the other hand, for I, J ∈ J (P) with I ≁ J and for ε, µ ∈ {−1, 0, 1} n , if ε p = µ p for any p ∈ max(I) ∩ max(J), then x ε I x µ J ∈ in(G O (e) ). Hence u and v are of the form where I k , J k ∈ J (P) and ε (k) = (ε (k) n ) ∈ {−1, 0, 1} n for k = 1, 2, . . ., r such that (a) I 1 ⊂ · · · ⊂ I r and J 1 ⊂ · · · ⊂ J r ; (b) For any p and q, and for any i ∈ max(I p ) ∩ max(I q ), we obtain ε i ; (c) For any p and q, and for any j ∈ max(J p ) ∩ max(J q ), we obtain µ , it then follows that max(I r ) = max(J r ) and ε r = µ r . Hence one has x (ε r ) J r . This contradicts the assumption that f is irreducible.
By Lemma 3.1 and Theorem 5.2, we have the following immediately. where I, J ∈ J (P), ε p = µ p , and p ∈ max(I) ∩ max(J) together with all monomials where I, J ∈ J (P) with I ≁ J and ε p = ε ′ p for each p ∈ max(I) ∩ max(J). Moreover, from Proposition 3.2, in < C (e) (I Let P be a finite poset on [n]. Given elements i, j of P, we say that j covers i if i < j and there exists no k ∈ P such that i < k < j. If j covers i in P, then we write i ⋖ j. A chain of P is a totally ordered subset of P. A chain of the form i 1 ⋖ i 2 ⋖ · · · ⋖ i r is called a saturated chain. A saturated chain i 1 ⋖i 2 ⋖· · · ⋖i r is called maximal if i 1 ∈ min(P) and i r ∈ max(P). First, we give the facet representations of enriched chain polytopes which easily follows from [16, Lemma 1.1] and the facet representations of chain polytopes [18]. where i 1 ⋖ i 2 ⋖ · · · ⋖ i r is a maximal chain of P, and ε j ∈ {1, −1}. In addition, each of the above inequalities is facet defining.
On the other hand, the facet representations of enriched order polytopes are as follows.
r is a maximal chain of P. In addition, each of the above inequalities is facet defining.
Proof. The proof is induction on n. If n = 1, then the assertion is trivial. Assume n ≥ 2.
Let Q ⊂ R n be the solution set of the above linear inequalities. Since 2 s−1 −∑ s j=2 2 s− j = 1 holds for any positive integer s, it is easy to see that e ε F min + e F comin satisfies (a) and (b) for any filter F of P, and for any ε ∈ {−1, 1} |F min | . Since O . , x n ) ∈ Q. First, we will show that |x i | ≤ 1 for each i ∈ [n]. Let i = i 1 ⋖ i 2 ⋖ · · · ⋖ i r be a saturated chain of P with i r ∈ max(P). Then x satisfies the following r inequalities: . . .
We now prove that x belongs to O (e) P by induction on n. Suppose that x i = 0 for some i ∈ min(P). Then (x 1 , . . . , x i−1 , x i+1 , . . . , x n ) ∈ R n−1 satisfies inequalities (a) and (b) for the subposet P \ {i} of P. By the assumption of induction, (x 1 , . . ., P\{i} . It then follows that x belongs to O (e) P . Thus we may assume that x i = 0 for any i ∈ min(P). Let λ = min{|x i | : i ∈ min(P)}. Note that 0 < λ ≤ 1. Let where F = [n], and ε ∈ {−1, 1} |F min | corresponds to the sign of x i for each i ∈ min(P) = F min . We now show that the vector y satisfies Inequality (c). If either x i 1 > 0 or i 1 / ∈ min(P) holds, then 13 If x i 1 < 0 and i 1 ∈ min(P), then λ + x i 1 ≤ 0 and hence If λ = 1, then we have y = 0 by inequalities (c) and (d). Hence x = e ε F min +e F comin ∈ O (e) P . If λ = 1, then 1 1−λ y belongs to Q by inequalities (c) and (d). From the definition of λ , there exists i ∈ min(P) such that y i = 0. By the assumption of induction, 1 1−λ y belongs to O (e) P , and hence y belongs to Finally, we will prove that each of inequalities (a) and (b) is facet defining. Let where i 1 ⋖ i 2 ⋖ · · · ⋖ i r is a saturated chain of P with i r ∈ max(P), and let where i 1 ⋖ i 2 ⋖ · · · ⋖ i r is a maximal chain of P. It is enough to show that Let i 1 ⋖ i 2 ⋖ · · · ⋖ i r be a saturated chain of P with i r ∈ max(P). If min(P) = {i 1 }, then let i = i 1 . If min(P) = {i 1 }, then let i be an arbitrary element in min(P) \ {i 1 }. Note that, if min(P) = {i 1 }, then i 2 ⋖ i 3 ⋖ · · · ⋖ i r is a maximal chain of P \ {i}. Let is unimodularly equivalent to a facet of O On the other hand, for a maximal chain i 1 ⋖ i 2 ⋖ · · · ⋖ i r of P, let i = i 1 if min(P) = {i 1 }, and let i be an arbitrary element in min(P) \ {i 1 } otherwise. Note that, if min(P) = {i 1 }, then i 2 ⋖ i 3 ⋖ · · · ⋖ i r is a maximal chain of P \ {i}. Then is unimodularly equivalent to a facet of O Given a polytope P of dimension n, let f n−1 (P) be the number of the facets of P. It is known [7, Corollary 1.2] that f n−1 (O P ) ≤ f n−1 (C P ) for any poset P. Proof. The formulas of the number of facets follows from Proposition 6.1 and Theorem 6.2. Each maximal chain of P of length ℓ contains exactly ℓ + 1 saturated chains of P that contains a maximal element of P. Since ℓ + 2 ≤ 2 ℓ+1 for any integer ℓ ≥ 0, we have sc(P) + mc(P) ≤ ∑ n−1 ℓ=0 2 ℓ+1 mc ℓ (P). 15 In [8, Lemma 3.8], tight upper bounds for f n−1 (O P ) and f n−1 (C P ) are given. Given an integer n ≥ 2, let It is known [14, Theorem 1] that µ n is the maximum number of cliques possible in a graph with n vertices.
In addition, both upper bounds are tight.
Proof. The proof for C  P holds for n ≤ 4. Assume n ≥ 5. Let P be a poset on [n]. Let P 1 = P and let M 1 be the set of all maximal elements of P 1 . If P 1 is not an antichain, then let P 2 = P 1 \ M 1 and let M 2 be the set of all maximal elements of P 2 . In general, if P i is not an antichain, then P i+1 = P i \ M i and let M i+1 be the set of all maximal elements of P i+1 . By this procedure, we get a sequence of posets P 1 , . . . , P r such that P r is an antichain. Then we have f n−1 (O (e) P ) ≤ |M 1 | + |M 1 ||M 2 | + · · · + |M 1 ||M 2 | · · · |M r−1 | + 2|M 1 ||M 2 | · · ·|M r |. We show that if n = 3k + 2, for n ≥ 5. Suppose that m 1 , . . . , m r , where 1 ≤ r ≤ n, ∑ r j=1 m j = n, and 1 ≤ m i ∈ Z give the maximum value of (4). If m i < m i+1 for some i, then where (m ′ i , m ′ i+1 ) = (m i+1 , m i ) and m ′ k = m k if k / ∈ {i, i +1}. This is a contradiction. Hence we have m 1 ≥ m 2 ≥ · · · ≥ m r . If m 1 ≥ 4, then Hence 2m 1 m 2 · · · m r + where m ′ 0 = m 1 +1 2 , m ′ 1 = m 1 − m ′ 0 and m ′ k = m k if k / ∈ {0, 1}. This is a contradiction.
A poset that attains the maximum value is the ordinal sum A r ⊕ · · · ⊕ A 1 of antichains A 1 , . . . , A r such that |A i | = m i .
Finally, we discuss when the number of facets of O P is equal to that of C (e) P . By the argument in the proof of Corollary 6.3, each maximal chain of P of length ℓ must satisfy ℓ + 2 = 2 ℓ+1 , and hence ℓ = 0. Thus P is an antichain.