Two poset polytopes
Received Februaty
Two Poset Polytopes 0
Richard P. Stanley 0
0 Department of Mathematics, Massachusetts Institute of Technology , Cambridge, MA 02139 , USA
Two convex polytopes, called the orderpolytope d)(P) and chain polytope <~(P), are associated with a finite poset P. There is a close interplay between the combinatorial structure of P and the geometric structure of E~(P). For instance, the order polynomial fl(P, m) of P and Ehrhart polynomial i(~9(P),m) of O(P) are related by f~(P,m+l)=i(d)(P),m). A "transfer m a p " then allows us to transfer properties of O(P) to W(P). In particular, we transfer known inequalities involving linear extensions of P to some new inequalities.

Our aim is to investigate two convex polytopes associated with a finite partially
ordered set (poset) P. The first of these, which we call the "order polytope" and
denote by O(P), has been the subject of considerable scrutiny, both explicit and
implicit, Much of what we say about the order polytope will be essentially a
review of wellknown results, albeit ones scattered throughout the literature,
sometimes in a rather obscure form. The second polytope, called the "chain
polytope" and denoted i f ( P ) , seems never to have been previously considered per
se. It is a special case of the vertexpacking polytope of a graph (see Section 2) but
has many special properties not in general valid or meaningful for graphs. There
is a surprising connection (Section 3) between (P(P) and (~(P) which will allow
us to " t r a n s f e r " properties of O(P) over to r ( ( p ) .
Given the poset P = {x 1. . . . . x , } (where by standard abuse of notation we
identify p with its set of points), the set R e of all functions f : P, R is an
ndimensional real vector space with a scalar product defined by ( f , g ) =
~.~Ej ( x ) g ( x ) , which makes R e a Euclidean space. In particular, we can talk
about convex subsets of R p and their volumes, orthogonal projections, etc.
*Partiallysupportedby NSF Grant No. 8104855MCSand by a GuggenheimFellowship.
Definition 1.1. The order polytope d)(P) of the poset P is the subset of R p
defined by the conditions
0 < f ( x ) < 1,
f ( x ) < f ( y )
for all x e P,
i f x < y in P.
(
1
)
(
2
)
(1')
(2')
(3a)
(3b)
(3c)
N o t e that O ( P ) is a convex polytope since it is defined by linear inequalities
and is bounded because of (
1
). Clearly, because of (
2
), we can replace (
1
) by the
conditions
0 < f ( x ) ,
if x is a minimal element of P,
f ( x ) < 1,
if x is a maximal element of P.
f ( x ) < f ( y )
if y covers x in P.
By the transitivity of P, we can replace (
2
) by the equivalent conditions
Let o: P ~ {1..... n } be a linear extension (orderpreserving bijection) of P.
We identify a with the permutation Yl. . . . . Yn of the elements x 1..... x n of P
defined by o(y,) = i. All functions f ~ R e satisfying 0 < f ( Y l ) < "'" < f(Yn) < 1
belong to 0 ( P ) . These functions form an ndimensional simplex, so we conclude
dim t V ( P ) = n. It is easily seen that conditions (1') and (2') are independent, so
they define the facets [(n 1)dimensional faces] of 0 ( P ) . More precisely a facet
of O ( P ) consists of those f ~d~(P) satisfying exactly one of the following
conditions:
f ( x ) = 0,
f ( x ) = 1,
f ( x ) = f ( y ) ,
for some minimal x ~ P,
for some maximal x ~ P,
for some y covering x in P.
It is convenient to state the above conditions in a more uniform way. Let
be the poset obtained from P by adjoining a minimum element 0 and a maximum
element i. Define a polytope ~ ( P ) to be the set of functions g ~ R ~' satisfying
g(
0
) = 0,
g ( x ) <_ g ( y )
g ( i ) = 1,
if x _< y in ,b.
The linear map P: ~ ( P ) * O ( P ) obtained by restriction to P is clearly a bijection
and hence (since P is linear) defines a combinatorial equivalence of polytopes.
Thus by (
3
) a facet of O ( P ) consists of those g ~ ~ ( P ) satisfying g(x) = g(y) for
some fixed pair (x, y)~ for which y covers x in P. In particular, the number of
facets of O ( P ) or O ( P ) is the number c ( t ' ) of cover relations in P, or
equivalently c(P)+ a+ b, where P has a minimal elements and b maximal
elements.
We now wish to determine the entire facial structure of t~(P), or equivalently
of 0 ( P ) . Since every face is an intersection of facets, it follows that a face F,~ of
~ ( P ) corresponds to certain partitions ~r= {B 1. . . . . Bk} of P into nonempty
pairwise disjoint blocks, viz.,
F,~ = ( g ~ ~ ( P ) : g is constant on the blocks Bi of Ir }.
(
4
)
It r e m a i n s to d e t e r m i n e for which rr F,~ is a face, and which are the distinct faces
F., Call i r a face partition if F,, is a face of P. It is clear that if ~r is a face
partition, then ~r is connected, i.e., every block B of ~r is connected as an
(induced) s u b p o s e t of P. Call a partition ~r = { B 1. . . . . B k } closed if for any i 4= j
there is g ~ F~ such that g(Bi) ~ g(Bj). Every partition ~r has a unique
coarsening ~ for which ~ is closed and F , = F ~ . Moreover, if ~ r ~ [ B 1. . . . . Bk} is a
closed face partition then dim F~ = k  2 [since if (~~ B, and 1 ~ By then g ~ F~
satisfies g ( B i ) = 0 and g ( B j ) = 1]. H e n c e it remains to describe the closed face
partitions. This description was a p p a r e n t l y first explicitly observed by Geissinger
[6]. W e will state Geissinger's result below ( T h e o r e m 1.2) but will omit the rather
straightforward proof.
Define a b i n a r y relation < ~ on ~r by setting B, < Bj if x _< y for some x ~ B i
and y ~ By. Call ~r compatible if the transitive closure of < ~ is a partial order
(i.e., is antisymmetric). If ~r is c o m p a t i b l e then every block B of sv is convex; i.e.,
if x, z ~ B a n d x < y < z.. then y ~ B. The converse is false; e.g., let I ' be given
by Fig. 1. T h e partition into blocks 0, ad, bc, 1 is connected and convex, but not
compatible.
T h e o r e m 1.2. A partition of P is a closed face partition if and on~ if it is connected
and compatible. (In particular, the partition ~r into a single block P yields the empty
set F, = 0 , which we regard as a face.)
T h u s the lattice of faces 0 ( P ) [or (~(P)] is isomorphic to the lattice of
connected c o m p a t i b l e partitions of P, ordered by reverse refinement. F o r
instance, if P = { a, b } is a twoelement antichain, then d~(P) is a square and Fig. 2
depicts its face lattice (with (~ and ] written 0 and 1).
Define a filter (or dual order ideal, upset, or increasing subset) of P to be a
subset I of P such that if x ~ I and y > x, then y ~ I. Let XI: P ~ R denote the
OobI~
Oobi
Oobl
~
Oba!
Obol
characteristic function of I; i.e.,
x ~ ( x ) =
The following corollary is immediate from Theorem 1.2 and can also be easily
proved directly.
Corollary 1.3. The vertices of tP(P) are the characteristic functions XI of filters I
of P. In particular, the number of vertices of t~(P) is the number of filters of P.
The Chain Polytope
Let us define a second polytope associated with a poset P = {x 1..... x , }.
Definition 2.1. The chain polytope ~ ( P ) of the poset P is the subset of R e
defined by the conditions
0 < g ( x ) ,
for all x ~ P ,
g ( Y l ) + ' ' ' + g ( Y k )
<1,
for every chain yl < "'" < Yk ° f P "
(
5
)
(
6
)
Again it is clear that ~ ( P ) is a convex polytope. Since f~(P) contains the
ndimensional simplex { g ~ R P : g(x) >0 for all x ~ P and g ( x l ) + . . . + g ( x , )
< 1}, we have d i m ~ ( P ) = n. In view of (
5
) we can replace (
6
) by
g ( Y l ) + ' ' " + g(Yk) < 1,
for every maximal chain Yl < " " " < Yk of P.
( 6 ' )
Conditions (
5
) and (6') are easily seen to be independent and thus define the
facets of ~ ( P ) . In particular, the number of facets of ~ ( P ) is equal to n + m(P),
where re(P) is the number of maximal chains of P.
A description of the faces of ~ ( P ) analogous to Theorem 1.2 seems messy
and will not be pursued here. However, we do have a simple description of the
vertices analogous to Corollary 1.3. Define an antichain of P to be a subset A of
pairwise incomparable elements of P.
Theorem 2.2. The vertices of ~¢(P) are the characteristic functions XA of
antichains of P. In particular, the number of vertices of T ( P ) is equal to the number of
antichains of P.
Proof. Clearly each XA ~ ~¢(P)" Since 0 < g(x) < 1 for all g ~ ~ ( P ) and x ~ P,
it follows that XA is a vertex of i f ( P ) .
Conversely, suppose g ~ ~ ( P ) and g ~ XA for any antichain A of P. Let
Q = ( x ~ P: 0 < g ( x ) <1}. Let Q1 be the set of minimal elements of Q and Q2
the set of minimal dements of Q  Q r One easily sees that since g ~ XA, Q1 and
Q2 are nonempty. Define
Define gl, g2: P ~ R by
= m i n { g ( x ) , l  g ( x ) : x ~ Q i u Q 2 }.
g , ( x ) =
( g ( x ) ,
x q~Qi u Q2
~ g ( x )  e ,
xEQ2,
( g ( x ) ,
gz(x) = ( g ( x )  e ,
[g(x)+e,
xq~PluO2
x~O1
x~Q2.
It is clear that gi, g2 E ~g(P). Since gl 4:g2 and g = ½(gi + g2), it follows that g
is not a vertex of ~ ( P ) . []
Theorem 2.2 is already known within a graphtheoretical context. Let G be a
graph (with no loops and multiple edges) on a vertex set V = ( x 1. . . . . x,,}. Let
~/'(G) _c R v denote the convex hull of the characteristic functions XA of
independent (stable) sets A of vertices; i.e., no two vertices in A are adjacent in G. Then
U ( G ) is called the vertexpackingpotytope of G. In particular, given a poset P
define its comparabilitygraph C o m ( P ) to be the graph whose vertices are the
elements of P, with x, y ~ P adjacent if x < y or y < x. Then an independent set
of vertices of C o m ( P ) is just an antichain A of P, so by Theorem 2.2 we have
U ( C o m ( P ) ) = c~(p). But since comparability graphs are perfect (e.g., [6, Thm.
5.34]) it follows from [2, Thin. 3.1] (or see [7, Thm. 3.14]) that the facets of
U ( C o m ( P ) ) are given by (
5
) and (6').
There is a wellknown bijection between filters I and antichains A of P, viz.,
I = ( y : y > x f o r s o m e x ~ A } ,
A = set of minimal dements of I.
Thus from Corollary 1.3 and Theorem 2.2 it follows that O ( P ) and ¢g(P) have
the same number of vertices. In general, however, O ( P ) and cg(p) need not have
the same number of /dimensional faces for i > 0 (and hence need not be
combinatorially equivalent). For instance, if P is given by Fig. 3, then O ( P ) has
eight facets and oK(p) has nine facets. There is, however, one class of posets for
which O ( P ) and cg(p) are in fact combinatorially equivalent.
Theorem 2.3. Suppose P has length at most one (i.e., P has no threeelement
chains). Then O( P ) and cg( p ) are affinely equivalent and hence combinatorially
equivalent.
Proof
Define a nonsingular affine transformation f: R e ___,R e by
( ~ f ) ( x ) = ( f ( x ) ,
l  f ( x ) ,
if x is a minimal element of P
otherwise
It is routine to check that the image of 0 ( P ) under ~ is ~ ( P ) , and the proof
follows. []
In Section 4 we generalize the fact that for any P, O(P) and ~ ( P ) have the
same number of vertices.
A Connection Between O(P) and ~ ( P )
In this section we construct a map q~: O ( P ) , g ( P ) with several nice properties.
This will allow us to transfer certain properties of 0 ( P ) over to i f ( P ) .
Definition 3.1. Let P be a finite poset, and define the transfer map ¢p: ¢ ( P ) *
~ ( P ) as follows: If f ~ 0 ( P ) and x ~ P then
(epf ) ( x ) = rnin{ f ( x )  f ( y ) : x covers y in P } .
(
7
)
Theorem 3.2. (a) The transfer map qJ is a continuous, piecewiselinear bijection
from O ( P ) onto cg(p).
(b) Let m be a positive integer and f ~ O( P). Then m f ( x ) ~ 7_ for all x ~ P if
and only if m ( q~f)( x ) ~ Z for all x ~ P.
Proof. (a) Continuity is immediate from the definition (
7
). Moreover, for each
linear extension Yl..... yn of P, q~is linear on the simplex defined by 0 < f ( Y l )
<• .. < f ( y , ) < 1 . Since these simplices dearly cover 0 ( P ) , it follows that q~ is
piecewiselinear. Now define q~: T ( P ) ,0 ( P ) by
( ~ g ) ( x )
= m a x { g ( y l ) +   . + g ( y k ) : Yt < ' ' "
< Y* = X } "
One checks that ( q ~ f f ) f = f and ( ~ q , ) g = g for all f ~ 0 ( P )
Hence ~ is a bijection (with inverse ~k).
(b) This result is immediate from (
7
) and (
8
).
and g ~ ( P ) .
(
8
)
[]
The Ehrhart Polynomial
Let ~ be a ddimensional convex polytope in R n with integer vertices. If m is a
positive integer then define
i ( ~ , m ) = c a r d ( m ~ n Z " ) .
In other words, i ( ~ , m) is equal to the number of points a ~ ~ such that
mct~ Z ' . It is known that i ( ~ , m) is a polynomial function of m of degree d,
called the Ehrhart polynomial of ~ . When d = n the leading coefficient of
i ( ~ , m) is the volume V ( ~ ) of ~ . For these and other facts concerning i ( ~ , m),
see, e.g., [15].
Now let P be a finite nelement poset and m a positive integer and define
f l ( P , m) to be the number of orderpreserving maps ,/: P ~ (1 ..... m ) ; i.e., if
x _< y in P then 71(x) < */(y). Then f~(P, m) is a polynomial function of m of
degree n, called the order polynomial of P. The leading coefficient of ~2(P, m) is
e ( P ) / n ! , where e ( P ) is the number of linear extensions of P. For these and
other facts concerning ~2(P, m), see, e.g., [12] and [13, Sections 13 and 19].
Theorem 4.1.
The Ehrhart polynomials of (9(P ) and ~ ( P ) are given by
i ( ( 9 ( P ) , m ) = i ( ~ ( P ) , m )
= ~ ( P , m + l ) .
Proof By definition, i(O(P), m) is equal to the number of orderpreserving
maps f : P ,R satisfying 0 < f ( x ) < 1 and m f ( x ) ~ Z for all x ~ P. This is
equivalent to the condition that mf: P ~ (0,1 . . . . . m} is orderpreserving, so
i(O(P), m) = ~2(P, m +1). But Theorem 3.2(b) implies that i(O(P), m) =
i ( ~ ( P ) , m), and the proof follows. []
Since the leading coefficient of i ( ~ , m ) is V ( ~ ) (when d i m # = n and
c R n) and that of ~2(P, m + 1 ) is e ( P ) / n ! , there follows
Corollary 4.2.
The volumes of O( P ) and cg( p ) are given by
V ( O ( P ) )
= V ( ~ ( P ) )
= e(P)/n!.
It would be interesting to find other vertexpacking polytopes whose volumes
have a simple combinatorial interpretation. Let us also mention that a method
similar to the proof of Corollary 4.2 for showing that two convex polytopes have
the same volume appears in [14].
Example 4.3. Let F, denote the nelement fence, i.e., the poset with elements
x 1. . . . . x , and cover relations
x, < Xi+l , i f i i s o d d ,
x, > xi+l, i f i i s e v e n .
A bijection o: F, ~ (1 ..... n } is orderpreserving if and only if the permutation
o ( x l ) , o ( x 2 ) . . . . . o ( x , ) of (1 . . . . . n} is alternating, i.e., o ( x l ) < o(x2) > o(x3) <
• . • . Hence e ( F , ) is the number E , of alternating permutations of (1 ..... n }. E ,
is an Euler number and is wellknown (e.g., [3, pp. 258259]) to satisfy
E n x n
Y'. ~
n>_0
= secx + t a n x .
T h e chain polytope ff(Fn) may be identified with the set of all vectors (Yl . . . . . Y,)
R" satisfying
(
9
)
(10)
Yi > 0,
1 < i < n, and
y~ + y~+x < 1,
l < i < n  1 .
V,x" = secx + tan x.
It follows from Corollary 4.2 that the volume Vn of the set (
9
) satisfies
Equation (10) was first given in [10] (see also [4]).
With almost no effort we obtain the following interesting corollary of
T h e o r e m 4.1.
Corollary 4.4. The order polynomial f~(P, m) of a finite poset P depends only on
the comparability graph C o m ( P ) of P.
Proof. By Theorem 4.1 we have f ~ ( P , m + l ) = i ( ~ ( P ) , m ) , and by definition
~ ( P ) depends only on C o m ( P ) . []
In particular, the leading coefficient of f~(P, m) depends only on C o m ( P ) ,
and we obtain
Corollary 4.5. The number e ( P ) of finear extension of P depends only on
C o m ( P ) .
Corollary 4.5 was first stated in [7, p. 139]. Its proof was based on a condition
as to when C o m ( P ) = Com(Q). This condition appears to be implicit in the work
of Gallai and others, but was apparently first explicitly stated in [5], and is given
as follows: Suppose P contains a poset P ' such that for all x ~ P  P', either (a)
x < y for all y ~ P', (b) x > y for all y ~ P', or (c) x and y are incomparable for
all y ~ P'. Define P: to be the poset obtained from P by dualizing P'; i.e., x < y
in P1 if and only if either (a) not both x ~ P ' and y ~ P ' , and x < y in P, or (b)
x and y ~ P ' and x > y in P. Call P1 a simple transform of P. Then C o m ( P ) =
C o m ( Q ) if and only if there is a sequence P = P0, P1 . . . . . Pk = Q of posets such
that each P~+I is a simple transform of Pg. It is then easy to check that simple
transforms have the same number of linear extensions, so Corollary 4.5 follows.
In fact, it is just as easy to check that simple transforms have the same order
polynomials, so Corollary 4.4 also follows. For another proof of Corollary 4.5 and
additional references, see [8].
N o t e that the proof we gave of Corollary 4.4 really has nothing to do with
convex polytopes. To see this, define for m > 1 the chain polynomial F ( P , m) to
be the number of maps g: P o {0,1,2 .... } such that g ( y l ) + .  . + g(Yk) < m  1
for all chains y: < . . . < Yk of P. Then (
7
) defines a bijection between
orderpreserving maps f : P ~ {0. . . . . m  1 } and maps q~f: P  o {0,1 .... } enumerated by
F ( P , m). Hence f~(P, m) = r ( P , m). But F ( P , m) depends only on C o m ( P ) , so
the same is true for f~(P, m).
Two PosetPolytopes
Of course Corollary 4.4 may be extended to the statement that any invariant
of P which can be computed in terms of cg(p) depends only on Com(P). In
Corollary 6.3 we will see another example of such an invariant.
Corollary 4.4 and its proof suggest that the combinatorial type of 0 ( P ) itself
may depend only on Com(P). However, if P is given by Fig. 3 then there is easily
seen to be a poset Q satisfying C o m ( P ) = Corn(Q) such that O(Q) has nine
facets, while ¢9(P) has eight facets.
5. T r i a n g u l a t i o n s
The polytope ¢ ( P ) has a canonical triangulation which can be transferred to
i f ( P ) . We describe this procedure in this section and give an application in the
next.
An order ideal of P is a subset I of P such that if x ~ I and y _<x, then
y ~ I. Let J ( P ) denote the poset (actually a distributive lattice) of order ideals of
P, ordered by inclusion. Let
K : I ~ c I 2 c . "
c l k
be a chain in J ( P ) (where/,1 c I i means t h a t / i  t is strictly contained in //).
Define a set FK ~ R e by
FK = ( f ~ R e: (a) f is constant on the subsets
I1, 12  I1. . . . . Ik  Ik1' P  Ik of P, and
(b) 0 = f ( I 1 ) <f(1211)< " " <f(PIk) = 1 ) .
Then F~: is a (k  1)dimensional simplex contained in 0 ( P ) , and the set { FK: K
is a chain of J ( P ) } is a triangulation A ( P ) of O(P). [The empty chain K
corresponds to the empty face of A(P).] In particular, the facets (maximal faces)
of A ( P ) are given by
0 < f ( y , ) < . . . < f ( y , ) ~ 1,
(
11
)
where y~.... , y, is a linear extension of P. The number of facets is e ( P ) and each
has volume 1 / n !, giving another proof that V( O( P )) = e ( P )/ n !.
For any poset Q define the order complex A(Q) [1, Section 3] to be the
abstract simplicial complex on Q whose faces are the chains of Q. Hence, as an
abstract simplicial complex, A ( P ) is isomorphic to A ( J ( p ) ) . In particular, the
geometric realization [A(J(p))[ of A ( J ( p ) ) is an ncell, a result which also
follows from very general considerations [11, Corollary 3.4.3] but here is
explained more concretely.
It follows from the definition (
7
) of the transfer map ~ that q~ is linear on
each face Fx of A ( P ) . Hence q~(FK) is a simplex, and (since q~ is continuous) the
set {q~(FK): Fx ~ A(P)} is a triangulation q~F(e) of oK(p). By applying q~to the
facet (
11
) of A ( P ) , an explicit description of the facets of q,A(P) can be deduced.
Namely, given a linear extension o: P* {1. . . . . n} with o ( y i ) = i and given
I
Kv, = Kk: zj < zj_ 1 < . . . < z o = Yk
(
12
)
inductively by the conditions that (a) z o = Yk; (b) among all z covered by z,, a ( z )
is maximized when z = z~+l; and (c) zj is a minimal element of P. Let F~ be the
facet (
11
) of A ( P ) ; i.e., Fo= FK, where K is the maximal chain q~c (Yl} c
{ Y l , Y2} c . . . c P of J ( P ) . Then the equations defining the facet q~(Fo) of
4~A(P) are given by
0 <_ f ( Y l ) ,
Y'~ f ( x )
x E K I
<_ ~., f ( x ) ,
X E~ Kt+ I
E f ( x ) <_1.
x ~ K,,
l < i < n  1 ,
For instance, let P and o be given by Fig. 4, where the element y, of P is
labeled i. Then, writing f~ for f(Yi), the equations for q,(Fo) are given by
0 < f t < f z < f 2 + f 3
< f z + f 3 + f , < f z + f 5
_<A+A+A_<I,
which may also be written as
o<A<A,
A + A  < A ,
o<A,
A + A + A  < I .
o<A,
o<A,
In [16] the AlexandrovFenchel inequalities from the theory of mixed volumes
were used to prove the logarithmic concavity of certain integer sequences
associated with O ( P ) . After reviewing this result we "'transfer" it to o f ( p ) and obtain
new logconcave sequences involving linear extensions of P.
We state the AlexandrovFenchel inequalities in a form most convenient for
our purposes. For references to their proofs, see [16]. Let (Ha: 0 < h _<1} be a
collection of parallel (affine) hyperplanes in R " such that the distance between
H x and H~, is 1~/~1. Let ~ 0 C H o and ~ l c H 1 be convex bodies (i.e.,
nonempty compact convex sets), and let ~ = c x ( ~ 0 W~1), the convex hull of ~ 0
and :~1 Set ~ x = :~ n H x and let V "  I ( ~ x ) denote the ( n  1 )  d i m e n s i o n a l
volume of ~ x  Then there exist real numbers V~(~0' ~ l ) > 0, 0 < i < n  1, such
that
V . _ ~ ( ~ x ) =
n
1 V(bao, ~1)~,(1  )k) "  1  / ,
0 _< X _< 1. (
13
)
The number V, = V,(~o, ~ t ) is called the ith m i x e d volume of ~ o and ~ l [in
particular, Vo(~o, ~al) = v "  l ( ~ o ) , V.l(~O, ~ 1 ) = v "  l ( ~ a l ) ] , and the
AlexandrovFenchel inequalities assert that
i=0
1~,2 > Vi_IV/+ 1,
Then the Oxs satisfy the conditions for (
13
). Moreover, if o is a linear extension
of P and F o the corresponding facet (
11
) of A ( P ) with x = y,, then Ox ~ Fo is
given by all f ~ R v satisfying
0 < f ( y , )
<_ . . . <_ f ( Y i ) = ~ < f ( Y i + l ) <<"'" < f ( Y , )
(
14
)
(
15
)
(
16
)
It follows that
and hence
vnl(¢
~ i  , ( 1 _ X) "  i
n F°) = ( i  1 ) ! ( n  i ) ! "
1 . ,
g n  l ( ~ ) h ) = ( n   1 ) !
( : )
E Na+l H . 1 ~ i ( 1  ) k ) n  l  i ,
i=O
N,+ 1 = (n 1)lV,(tV0, ~1),
where Nj is the number of linear extensions o of P satisfying o ( x ) = j . Therefore
and we conclude from (
13
) that Ni 2 > Ni_INt+I, 2 < i < n  1. More details are
given in [16, Section 3] in a somewhat more general setting.
We n o w wish to "transfer" (
15
) and (
16
) to the chain polytope @ = @(P).
We cannot simply define cgx = ~tVx, since ~O x need not lie in a hyperplane.
Rather, we define @x in analogy to o u r definition of • x and compute V(gx) by
examining each tpq~xN Fo, where ff is given by (
8
). Thus fix x ~ P, and for
0 < ) ~ < 1 set
% =
f ( x ) = x ) .
L e m m a 6.1.
~ ( P ) = c x ( ~ o U ~¢1).
Proof. By Theorem 2.2 every vertex of ~ ( P ) lies in ~0 u ~i, and the proof
follows. []
It follows that there are numbers Mo, M I. . . . . 3//._ 1 (depending on the choice
of x ~ P ) uniquely defined by
M i ( n s t l ) h i ( 1  h ) ~  i  ' ,
0 < ~ < _ _ 1,
(17)
and that then Mi2 >__Mi_xM~+ 1, 1 < i < n  2 . It remains to interpret M~
combinatorially.
Theorem 6.2. M i is equal to the number of linear extensions o: P >{1,..., n )
such that if o ( x ) = s, then i is the largest integer (necessarily less than s) for which
o  l( s  1), o  l( s  2 ) . . . . . o  l( s  i) are all incomparable with x. (In particular,
i = 0 i f s = 1 or if o  l ( s  1 ) < o  l ( s ) in P. I f x is a minimal element of P then
i = s  1 . )
Proof. Since the simplices ~Fo are the facets of the triangulation ~ A ( P ) of
~ ( P ) , we have
V "  l ( f f x ) = Z V"X(ffx n epFo),
o
(18)
summed over all linear extensions o of P. Define a map p: RP,R p',x) by
restricting f ~ R e to P  ( x }. Since p is a projection orthogonal to ~x, we have
V "  l(cgx n ePFo) = V( P(ffx n ~Fo)), where Vdenotes ordinary (n  1)dimensional
volume (Lebesgue measure) in R e  ( x )  R "  i . Let ~: i f ( P ) , 0 ( P ) be the
bijection defined by (
8
). Consider the composition p + : ~x >R e  ( x ) . From (
8
) it
follows that for any y ~ P  {x } and any g ~ fix n epFo, we have
( p ~ G ) ( y )
=
~_, g ( z ) ,
z E K~v
where K v is the chain (
12
) defined in Section 5 (and where g ( x ) = ~ by the
definition of ~x). Hence the map pff, when restricted to ~x n epFo, is an affine
transformation whose linear part can be put in triangular form with ls on the
diagonal. In particular, ptk is volumepreserving, so
v "  i ( p ~ ( ~ x N t k F o ) )
= V n  i ( p ( ~ x n F o )
) = V n  i ( ~ x n + F o ) .
(19)
T h e first equality holds because ~ke~= identity.
Let y be that d e m e n t of P covered by x which maximizes o ( y ) . Then the
condition g ( x ) = A for g ~ ~¢xn ~F, is by (
7
) equivalent to f ( x )  f ( y ) = ~ in
ff(~Cxn~Fo). [If x is minimal then the condition becomes f ( x ) = k.] Define
Yi = o  1 ( 0 and suppose y = y,, x = x,. Set f~ = f ( Y i ) . Then the set P f f ( ~ x n fiFo)
t~i
( r  l ) !
X,~I
( s  r  l ) !
( l _ X _ t ) .  ~
(n s)!
v , ,  l ( p ~ ( ~ n , r o ) ) = ( r  1 ) ! ( s
r  1 ) ! ( n
s)!
× £ 1  x t ~ _ l ( 1 _ ~ _ t ) "  ' d t .
Hence
Hence
Let t = u(1  X) to obtain
This latter integral is just the beta function
B ( r , n  s + l )
=
( r  1 ) ! ( n  s ) !
( n + r  s ) !
is defined by the conditions
O < f l <
. "
<_L_I<_L+x<_L+I<_ . . . _< f , _ < l .
For fixed fr = t (where 0 _<t < 1  X) the projection of (20) orthogonal to the
plane £ = t has (n 2)dimensional volume
f o l  X t '  l ( l   ~  t ) "  S d t = ( 1  ) ~ ) " + r  ~ f o l u ~  l ( 1  u)nSdu.
Xsrl(l__X)"+r'
V"l(p+ ( % n , r o ) ) = ( s  r  1 ) ! ( n + r  s ) ! "
Set s  r  1 = h(o) = h(o, x). Comparing (18), (19), and (21) yields
(n  1)!V"1(%ax) = ~ (nh(_o )1) )~h(o)(1
X)nlh(o).
But clearly h ( o ) is just the largest integer i for which o  l ( s  1 ) , o  l ( s
2)..... o  l ( s  i) are all incomparable with x, and the proof follows by
comparing (17) and (22). []
Example 6.3. Let P be given by Fig. 5, with x, labeled i. Choose x = x 4. We list
the linear extensions of P, with the elements o  l ( s  1 ) ..... o  l ( s  i)
incom(20)
(21)
(22)
parable with x underlined:
1 2 3 4 5
2 1 3 4 5
1 2 4 3 5
2 1 4 3 5
2 4 1 3 5
1 2 4 5 3
2 1 4 5 3
2 4 1 5 3
2 4 5 1 3
Hence M0 = 4 , M 1 = 3 , M 2 = l , M 3 = M 4 = 0 .
Since ~ ( P ) depends only on Corn(P) we obtain, just as for Corollary 4.4, the
following corollary.
Corollary 6.4. For any nelement poset P a n d any x ~ P, the numbers
M o , M 1. . . . , M ~ _ 1defined in Theorem 6.2 depend only on Corn(P) with the vertex x
specified.
It is not even a priori obvious that the Mrs are unaffected by replacing P
with its dual P* (and leaving the choice of x unaltered), but a simple
combinatorial proof which we omit can be given. More generally, Corollary 6.4 can also
be proved using the result of Gallai et al. discussed after Corollary 4.5.
Just as Theorem 6.2 is the " ~ ( P ) analogue" of (
16
), so Theorem 6.2 and its
consequence M i 2 > M i _ l M i + l can be straightforwardly generalized to give a
~ ( P ) analogue of the generalization of (
16
) given in [16, Thin. 3.2]. Moreover, a
variation of (
15
) given in [9, (2.14)] can also be given a c~(p) analogue. We will
n o t enter into details here.
A general property of the mixed volumes V/of (
13
) asserts that if V, = 0 then
either Vo = V 1 . . . . . V; = 0 or Vi = V~+x. . . . . IT,_ 1 = O. This property,
together with (
14
) and the fact that V, > 0, implies that the sequence V0, V1,..., V,_ 1
is unimodal; i.e., for some j we have Vo < V 1 < . . . < Vj > Vj+ 1 > . . . > V , _ p In
the case of the numbers Mi of Theorem 6.2, an even stronger result can be proved
b y direct combinatorial reasoning.
Theorem 6.5. The numbers M i = Mi( x ) o f Theorem 6.2 are weakly decreasing;
i.e., M o > M I > . . . > Mn_ 1.
Proof L e t f~i b e the set o f linear extensions o o f P satisfying h ( a ) = i, where
h ( o ) is as i n (22). I f i > 0 a n d o ~ fli,then define o ' : P , {1 . . . . . n } b y
~i i n t o f~i1. Since If~,l = M i, the p r o o f
[]
1. A. Bj~Srner , A. Garsia , and R. Stanley , An introduction to the theory of CohenMacaulay partially ordered sets, in Ordered Sets (I . Rival, ed.), Reidel, DordrechtBostonLondon, 1982 , 583  616 .
2. V. Chv~tal, On certain pdytopes associated with graphs , J. Combin. Theory Ser B 18 ( 1975 ), 138  154 .
3. L. Comtet , Advanced Combinatorics, Reidel, DordrechtBoston, 1974 .
4. E.E. Doberkat , Problem 84 20, SIAM Review 26 ( 1984 ), 580 .
5. B. Dreesen , W. Poguntke , and P. Winkler , Comparability invariance of the fixed point property, preprint .
6. U Geissinger , A polytope associated to a finite ordered set, preprint .
7. M . C . Golumbic , Algorithmic Graph Theory and Perfect Graphs , Academic Press, New York, 1980 .
8. M. Habib , Comparability invariants, Ann. Discrete Math. 23 ( 1984 ), 371  386 .
9. J. Kahn and M. Saks , Balancing poset extensions, Order 1 ( 1984 ), 113  126 .
t0. I.G. Macdonald and R. B. Nelsen (independently), Solution to E2701, Amer. Math. Monthly 86 ( 1979 ), 396 .
11. J. S. Provan and U J. Billera , Simplicial Complexes Associated with Convex Polyhedra, I: Constructions and Combinatorial Examples , Technical Rept. no. 402 , School of Operations Research and Industrial Engineering, Comell University, Ithaca, New York, January 1979 .
12. R. Stanley , A chromaticlike polynomial for ordered sets ., in Proc. Second Chapel Hill conference on Combinatorial Mathematics and Its Applications (May , 1970 ), Univ. of North Carolina, Chapel Hill, 421  427 .
13. R. Stanley , Ordered structures and partitions , Mem. Amer. Math. Society, no. 119 , 1972 .
14. R. Stanley , Eulerian partitions of a unit hypercube, in Higher Combinatorics (M. Aigner , ed.), Reidel, DordrechtBoston, 1977 , p. 49 .
15. R. Stanley , Decompositions of rational convex polytopes , Ann. Discrete Math. 6 ( 1980 ), 333  342 .
16. R. Stanley , Two combinatorial applications of the AleksandrovFenchel inequalities , J. Combin. Theory Ser A 31 ( 1981 ), 56  65 .