Flag enumerations of matroid base polytopes

0.22 MB·

by  Sangwook Kim

FLAG ENUMERATIONS OF MATROID BASE POLYTOPES
SANGWOOK KIM
9
0
0 Abstract. Inthis paper,westudyflagstructuresofmatroidbasepoly-
2 topes. We describe faces of matroid base polytopes in terms of matroid

n data,andgiveconditionsforhyperplanesplitsofmatroidbasepolytopes.
a Also, we show how the cd-index of a polytope can be expressed when a
J polytope is split by a hyperplane, and apply these to the cd-index of a

0 matroid base polytope of a rank 2 matroid.
3

]
O
C 1. Introduction
.
h For a matroid M on [n], a matroid base polytope Q(M) is the polytope in
t
a Rn whose vertices are the incidence vectors of the bases of M. The polytope
m
Q(M) is a face of a matroid polytope first studied by Edmonds [6], whose
[
vertices are the incidence vectors of all independent sets in M. In this paper,

2 we study flags of faces and the cd-index of matroid base polytopes.
v
It is known that a face σ of a matroid base polytope is the matroid base
4
3 polytope Q(M ) for some matroid M on [n] (see [8] and Section 2 be-
σ σ
5
low). We show that M can be described using equivalence classes of factor-
3 σ
. connected flags of subsets of [n]. As a result, one can describe faces of Q(M)
1
in terms of matroid data:
0
9
0 Theorem 1.1 (Theorem 2.9). Let M be a connected matroid on a ground
: set [n]. For a face σ of the matroid base polytope Q(M), one can associate
v
i a poset P defined as follows:
X σ
(i) the elements of P are the connected components of M , and
r σ σ
a
(ii) for distinct connected components C and C of M , C < C if and
1 2 σ 1 2
only if
C ⊆ S ⊆ [n] and σ ⊆ H implies C ⊆ S,
2 S 1
where H is the hyperplane in Rn defined by x = r(S).
S e∈S e
The cd-index Ψ(Q) of a polytope Q, a polynomiaPl in the noncommutative
variables c and d, is a very compact encoding of the flag numbers of a
polytope Q [2]. Ehrenborg and Readdy [7] express the cd-indices of a prism,
a pyramid, and a bipyramid of a polytope Q in terms of cd-indices of Q
Key words and phrases. Matroid base polytopes, cd-index.
1
and its faces. Also, the cd-index of zonotopes, a special class of polytopes,
is well-understood [3, 4]. Generalizing the formulas of the cd-indices of a
prism and a pyramid of a polytope, we show how the cd-index of a polytope
can be expressed when a polytope is split by a hyperplane in Section 3.
In Section 4, we find the conditions when a matroid base polytope is split
into two matroid base polytopes by a hyperplane:
Theorem 1.2 (Theorem 4.1). Let M be a rank r matroid on [n] and H be
a hyperplane in Rn given by x = k. Then H decomposes Q(M) into
e∈S e
two matroid base polytopes if and only if
P
(i) r(S) > k and r(Sc) > r −k,
(ii) if I and I are k-element independent subsets of S such that (M/I )|
1 2 1 Sc
and (M/I )| have rank r −k, then (M/I )| = (M/I )| .
2 Sc 1 Sc 2 Sc
We apply this theorem to the cd-index of matroid base polytopes for rank
2 matroids in Section 5.
Throughout this paper, we assume familiarity with the basic concepts of
matroid theory. For further information, we refer the readers to [13].
2. Matroid base polytopes
This section contains the description of faces of matroid base polytopes.
In particular, we associate a poset to each face of a matroid base polytope.
We start with a precise characterization of matroid base polytopes. Let B
be a collection of r-element subsets of [n]. For each subset B = {b ,...,b },
1 r
let
e = e +···+e ∈ Rn,
B b1 br
wheree istheithstandardbasisvectorofRn. ThecollectionBisrepresented
i
by the convex hull of these points
Q(B) = conv{e : B ∈ B}.
B
This is a convex polytope of dimension ≤ n−1 and is a subset of the (n−1)-
simplex
∆ = {(x ,...,x ) ∈ Rn : x ≥ 0,...,x ≥ 0,x +···+x = r}.
n 1 n 1 n 1 n
Gelfand, Goresky, MacPherson, and Serganova [9, Thm. 4.1] show the fol-
lowing characterization of matroid base polytopes.
Theorem 2.1. B is the collection of bases of a matroid if and only if every
edge of the polytope Q(B) is parallel to a difference e − e of two distinct
α β
standard basis vectors.
2
For a rank r matroid M on a ground set [n] with a set of bases B(M), the
polytope Q(M) := Q(B(M)) is called the matroid base polytope of M.
By the definition, the vertices of Q(M) represent the bases of M. For two
bases B and B′ in B(M), eB and eB′ are connected by an edge if and only if
eB−eB′ = eα−eβ for some α,β ∈ [n]. Since the latter condition is equivalent
to B \B′ = {α} and B′ \B = {β}, the edges of Q(M) represent the basis
exchange axiom. The basis exchange axiom gives the following equivalence
relation on the ground set [n] of the matroid M: α and β are equivalent if
thereexist bases B andB′ inB(M) withB\B′ = {α}andB′\B = {β}. The
equivalence classes are called the connected components of M. The matroid
M is called connected if it has only one connected component. Feichtner and
Sturmfels [8, Prop. 2.4] express the dimension of the matroid base polytope
Q(M) in terms of the number of connected components of M.
Proposition 2.2. Let M be a matroid on [n]. The dimension of the matroid
base polytope Q(M) equals n−c(M), where c(M) is the number of connected
components of M.
Theorem 2.1 implies that every face of a matroid base polytope is also a
matroid base polytope. For a face σ of Q(M), let M denote the matroid on
σ
[n]whose matroidbasepolytopeisσ. Forω ∈ Rn, let M denotethematroid
ω
whosebasesB(M )isthecollectionofbasesofM having minimum ω-weight.
ω
n
Then Q(M ) is the face of Q(M) at which the linear form ω x attains
ω i=1 i i
its minimum. Let F(ω) denote the unique flag of subsets
P
{∅ =: S ⊆ S ⊆ ··· ⊆ S ⊆ S := [n]}
0 1 k k+1
for which ω is constant on each set S \S and ω| < ω| . Ardila
i i−1 Si\Si−1 Si+1\Si
and Klivans [1] show that M depends only on F(ω), and hence one can call
ω
it M . They also give the following description of M .
F F
Proposition 2.3. [1, Prop. 2] Let M be a matroid on [n] and F be a flag
of subsets
{∅ =: S ⊆ S ⊆ ··· ⊆ S ⊆ S := [n]},
0 1 k k+1
then
k+1
M = (M| )/S ,
F Si i−1
i=1
M
where M| is the restriction of M on S and M/S is the contraction of M
S
on S.
A flag F = {∅ =: S ⊆ S ⊆ ··· ⊆ S ⊆ S := [n]} is called factor-
0 1 k k+1
connected (with respect to M) if the matroids (M| )/S are connected for
Si i−1
all i = 1,...,k+1. If F is factor-connected, then the connected components
3
of M are S \S ,S \S ,...,S \S . Proposition 2.2 and Proposition 2.3
F 1 0 2 1 k+1 k
show that the dimension of Q(M ) is n−k−1 when F is factor-connected.
F
Since Q(M ⊕M ) = Q(M )×Q(M ), it is enough to restrict to attention
1 2 1 2
to connected matroid M. For a connected matroid M on [n], facets of Q(M)
correspond to factor-connected flags of the form {∅ ⊆ S ⊆ [n]}. Feichtner
and Sturmfels [8] show that there are two types of facets of Q(M):
(i) a facet corresponding to a factor-connected flag {∅ ⊆ F ⊆ [n]} for some
flat F of M (in this case, the facet is called a flacet),
(ii) a facet corresponding to a factor-connected flag {∅ ⊆ S ⊆ [n]} for an
(n−1)-subset S of [n].
Lemma 2.4. Let M be a connected matroid on [n] and
F = {∅ =: S ⊆ S ⊆ ··· ⊂ S ⊆ S := [n]}
0 1 k k+1
a factor-connected flag with respect to M. Then the matroid (M| )/S
Sj+1 j−1
has at most two connected components for 1 ≤ j ≤ k.
(i) If it has one connected component, the flag
G = {∅ =: S ⊆ ··· ⊆ S ⊆ S ⊆ ··· ⊆ S := [n]}
0 j−1 j+1 k+1
is factor-connected and Q(M ) covers Q(M ) in the face lattice of
G F
Q(M).
(ii) If it has two connected components, then they are S \S and S \S .
j j−1 j+1 j
Moreover, the flag
F′ = {∅ =: S ⊆ ··· ⊆ S ⊆ S′ ⊆ S ⊆ ··· ⊆ S := [n]},
0 j−1 j j+1 k+1
whereSj′ = Sj−1∪(Sj+1\Sj), is factor-connectedandQ(MF′) = Q(MF).
In this case, F and F′ are said to be adjacent.
Proof. For j = 1,...,k, we have (M| )/S = [(M| )/S ]/(S \S )
Sj+1 j Sj+1 j−1 j j−1
and (M| )/S = [(M| )/S ]| . The first assertion is obtained from
Sj j−1 Sj+1 j−1 Sj
[13, Proposition 4.2.10], and the other assertions follow from [13, Proposition
(cid:3)
4.2.13] and Proposition 2.3.
Two factor-connected flags F and F′ of the same length are said to be
equivalent if there is a sequence of factor-connected flags
F = F ,F ,...,F = F′
0 1 r
such that F is adjacent to F for i = 1,...,r. We write F ∼ F′ when
i i−1
factor-connected flags F and F′ are equivalent.
Lemma 2.5. Let M be a connected matroid on [n] and X,Y disjoint subsets
of [n] with X ∪Y = [n]. Then B((M| )⊕(M/X)) and B((M| )⊕(M/Y))
X Y
are disjoint.
4
Proof. Since X ∩ Y = ∅ and X ∪ Y = [n], one has r(X) + r(Y) ≥ r(M).
If r(X) + r(Y) = r(M), then |B ∩ X| = r(X) and |B ∩ Y| = r(Y) for all
B ∈ B(M), andhence M = M| ⊕M| . But then M isnot connected. Thus
X Y
r(X)+r(Y) > r(M). Therefore there is no base which has r(X) elements in
X and r(Y) elements in Y, i.e., B((M| )⊕(M/X)) and B((M| )⊕(M/Y))
X Y
(cid:3)
are disjoint.
The following proposition shows that the equivalence classes of factor-
connected flags characterize faces of a matroid base polytope.
Proposition 2.6. Let M be a connected matroid on [n]. If F and F′ are
two factor-connected flags of subsets of [n] given by
F ={∅ =: S ⊆ S ⊆ ··· ⊆ S ⊆ S := [n]},
0 1 k k+1
F′ ={∅ =: T ⊆ T ⊆ ··· ⊆ T ⊆ T := [n]},
0 1 l l+1
then MF = MF′ if and only if F and F′ are equivalent.
Proof. If F ∼ F′, then MF = MF′ from Lemma 2.4.
For the other direction, suppose that MF = MF′. Then F and F′ have
the same length since dimQ(MF) = n−k−1 and dimQ(MF′) = n−l−1.
Without loss of generality, we may assume that S 6= T and S 6= T .
1 1 k k
We will use induction on k. We claim that
(2.1) T = S \S for some m > 1.
1 m m−1
This claim is going to be used both in the base case and the inductive step.
Let m be the smallest index such that T ∩ S 6= ∅ and let α ∈ T ∩ S .
1 m 1 m
Suppose that T is not contained in S , i.e., there is an element β ∈ T \S .
1 m 1 m
Since M| is connected, there are B′,B′ ∈ B(M| ) such that α ∈ B′ and
T1 1 1 T1 1
B′ = (B′ \{α}) ∪{β}. Choose B′ ∈ B((M| )/T ) for j = 2,...,k + 1.
1 1 j Tj j−1
Then B′ = B′ ∪B′ ∪···∪B′ and B′e= B′ ∪B′ ∪···∪B′ are bases of
1 2 k+1 1 2 k+1
e
MF′. But either B′ or B′ is not a base of MF since
e e
|B′ ∩(S \S )| = |B′ ∩(S \S )|−1.
m m−1 m m−1
e
This contradicts the assumption MF = MF′. Therefore T1 ⊆ Sm.
e
Now suppose (S \ S ) \ T 6= ∅, i.e., there is γ ∈ (S \ S ) \ T .
m m−1 1 m m−1 1
Since (M| )/S is connected, there are bases B ,B of (M| )/S
Sm m−1 m m Sm m−1
satisfying α ∈ B and B = (B \{α})∪{γ}. If B ∈ B((M| )/S ) for
m m m j Sj j−1
all j 6= m, then e
e
B = B ∪···∪B and
1 k+1
B = B ∪···∪B ∪B ∪B ∪···∪B
1 m−1 m m+1 k+1
5
e e
are contained in B(MF). But at least one of them is not in B(MF′) because
|B ∩T1| = |B ∩T1|−1. This is impossible because MF = MF′. Therefore
T = S \S .
1 m m−1
eBase case: k = 1. Equation (2.1) shows that T = S \S . Since T ∪S =
1 2 1 1 1
[n], M is not connected by Lemma 2.5. Thus, F and F′ are equivalent by
Lemma 2.4(ii).
Inductive step. Now suppose k > 1. By Equation (2.1), T = S \S
1 m m−1
for some m > 1. Let F be a flag
∅ =: S ⊆ S ⊆ S ∪T ⊆ S ∪T ⊆ ··· ⊆ S ∪T ⊆ S ··· ⊆ S := [n].
0 1 1 1 2 1 m−1 1 m+1 k+1
e
=Sm
WeclaimthatM| = M| ⊕M| forj = 1,...,m−1. LetB ∈ B(M| )
Sj∪T1 Sj T1 | {z } j Sj
and B′ ∈ B(M| ). Since
1 T1
MF = ⊕ki=+11(M|Si)/Si−1 = ⊕ki=+11(M|Ti)/Ti−1 = MF′
and M| = (M| )/S , there is a base B ∈ B(M ) such that B∩S = B
T1 Sm m−1 F j j
and B ∩T = B′. Thus B ∪B′ ∈ B(M| ), and hence
1 1 j 1 Sj∪T1
r(M| ) = r(M| )+r(M| ).
Sj∪T1 Sj T1
By [12, Lemma 3], we have M| = M| ⊕M| . Therefore
Sj∪T1 Sj T1
(M| )/S = [M| ⊕M| ]/S
S1∪T1 1 T1 S1 1
= (M| )/S
Sm m−1
and
(M| )/(S ∪T ) = [M| ⊕M| ]/(S ∪T ))
Si∪T1 i−1 1 T1 Si i−1 1
= (M| )/S
Si i−1
for i = 2,...,m − 1. Thus F is factor-connected and MFe = MF. By the
induction assumption, F ∼ F. Similarly, one can show S = T \ T for
1 l l−1
some l > 1 and the chain F′ geiven by
e
∅ =: T ⊆ T ⊆ S ∪T ⊆ S ∪T ⊆ ··· ⊆ S ∪T ⊆ T ··· ⊆ T := [n]
0 1 1 1 1 2 1 l−1 l+1 k+1
e
=Tl
is factor-connected and MFe′ = MF′. Thu|s th{ezind}uction assumption gives
F′ ∼ F′.
By the induction assumption again, we have F ∼ F′ and hence F and F′
(cid:3)
aere equivalent.
e e
If M is a matroid on [n] and S is a subset of [n], then the hyperplane
H defined by x = r(S) is a supporting hyperplane of Q(M) and
S e∈S e
Q(M) ∩ H is the matroid base polytope for (M| ) ⊕ (M/S). The next
S S
P
lemma tells us when a face of Q(M) is contained in H .
S
6
Lemma 2.7. Let M be a matroid on [n] and S be a subset of [n]. A face σ
of Q(M) is contained in H if and only if there is a factor-connected flag
S
F = {∅ =: S ⊆ S ⊆ ··· ⊆ S ⊆ S := [n]}
0 1 k k+1
containing S and σ = Q(M ).
F
Proof. If there is a flag F containing S and σ = Q(M ), then every base of
F
M is contained in B((M| )⊕(M/S)) and hence σ is contained in H .
F S S
For the converse, suppose σ is contained in H . Since σ is a face of Q(M),
S
there is a factor-connected flag
F′ = {∅ =: T ⊆ T ⊆ ··· ⊆ T ⊆ T := [n]},
0 1 k k+1
such that σ = Q(MF′) and k = n − dimσ − 1. We claim that Ti \ Ti−1
is contained in either S or [n] \ S for i = 1,...,k + 1. Suppose there are
α ∈ (T \T )∩S andβ ∈ (T \T )\S. Since(M| )/T isconnected, there
i i−1 i i−1 Ti i−1
exist bases B1 and B2 of MF′ with α ∈ B1 and B2 = (B1\{α})∪{β}. Then
|B ∩S| > |B ∩S| and so σ is not contained in H , which is a contradiction.
1 2 S
Construct aflag F asfollows: S = S ∪(T \T ), where l isthesmallest
j j−1 l l−1
index such that
(i) T \T ⊆ S \S if S \S 6= ∅,
l l−1 j−1 j−1
(ii) T \T ⊆ [n]\S if S \S = ∅.
l l−1 j−1 j−1
By the construction of F, S is contained in F. Since σ is contained in H ,
S
MF = ⊕ki=+11(M|Si)/Si−1 = ⊕ki=+11(M|Ti)/Ti−1 = MF′
and hence σ = Q(M ). (cid:3)
F
For a face σ of Q(M), let L be the poset of all subsets of [n] which
σ
are contained in some factor-connected flag F with σ = Q(M ) ordered by
F
inclusion. The following lemma shows that L is a lattice.
σ
Lemma 2.8. LetM be a matroidon [n] andS,T ⊆ [n]. IfQ(M) is contained
in H and H , then it is also contained in H .
S T S∩T
Proof. Since Q(M) is contained in H and H , we have
S T
r(S ∪T)+r(S ∩T) = r(S)+r(T)
(see [13, Lemma 1.3.1]). Suppose that Q(M) is not contained in H , i.e.,
S∩T
there is a base B of M such that |B ∩(S ∩T)| < r(S ∩T). Then
|B ∩(S ∪T)| = |B ∩S|+|B ∩T|−|B ∩(S ∩T)|
> r(S)+r(T)−r(S ∩T) = r(S ∪T)
which is impossible. Therefore Q(M) is contained in H . (cid:3)
S∩T
7
Since L is a sublattice of the Boolean lattice B , it is distributive. The
σ n
fundamental theorem for finite distributive lattices [17] shows that there is a
finite poset P for which L is the lattice of order ideals of P . The following
σ σ σ
theorem describes the poset P in terms of matroid data.
σ
Theorem 2.9. Let M be a matroid on [n] and σ be a face of Q(M). Then
L is the lattice of order ideals of P , where P is the poset defined as follows:
σ σ σ
(i) The elements of P are the connected components of M , and
σ σ
(ii) for distinct connected components C and C of M , C < C if and
1 2 σ 1 2
only if
C ⊆ S ⊆ [n] and σ ⊆ H implies C ⊆ S.
2 S 1
Note that P is a well-defined poset. Reflexivity and transitivity are clear.
σ
Suppose C and C are distinct connected components of M with C < C .
1 2 σ 1 2
Consider a minimal subset S such that C ⊆ S and σ ⊆ H . C < C implies
2 S 1 2
C is contained in S. Since C is a connected component of M , Lemma 2.7
1 2 σ
implies that there is a flag containing S and S \C such that Q(M ) = σ.
2 F
Since C ⊆ S \C , σ ⊆ H , and C * S \C , we have C ≮ C .
1 2 S\C2 2 2 2 1
Example 2.10. Let M be the rank 2 matroid on [4] = {1,2,3,4} whose
2,1,1
unique non-base is 12 and let σ be an edge of Q(M ) connecting e and
2,1,1 14
e . Then the connected components of M are 12, 3 and 4. Since {1,2,3,4}
24 σ
is the only subset S containing {3} such that σ ⊆ H , 12 < 3 and 4 < 3
S
in P . One can see that there are no other relations in P . Figure 1 is the
σ σ
proper part of the face poset of Q(M ) whose faces (shown on the right-
2,1,1
hand side of each box) are labeled by corresponding posets (shown on the
left-hand side of each box) and P is shown in the shaded box.
σ
Proof of Theorem 2.9. Let S be a set in L . By Lemma 2.7, there is a factor-
σ
connected flag
F = {∅ =: S ⊆ S ⊆ ··· ⊆ S ⊆ S := [n]}
0 1 k k+1
such that M = M and S = S for some i. Then S = ∪i (S \ S ).
σ F i j=1 j j−1
SupposeS \S < S \S inP forsomej ≤ i. Sinceσ ⊆ H , S \S ⊆ S
l l−1 j j−1 σ S l l−1
and hence S is an order ideal of P .
σ
Conversely, suppose T is an order ideal of P . Let U be the intersection
σ
of all subsets T such that σ ⊆ HTe and T ⊆ T. By Lemma 2.8, U lies in Lσ.
Suppose T 6= U. By Lemma 2.7, there is a factor-connected flag
e e
F = {∅ =: U ⊆ U ⊆ ··· ⊆ U ⊆ U := [n]}
0 1 k k+1
such that σ = Q(M ) and U = U for some j. Since T 6= U and T is an
F j
order ideal, we may choose F so that U \ U * T. Then U ∈ L and
j j−1 j−1 σ
T ⊆ U which is a contradiction. (cid:3)
j−1
8
2 34 123 34 124 34 34 23 24 1 34
134 13 14 4 14 24 3 13 23 12 13 14 234 23 24
2 34 2 34 14 24 1 34 23 1 34 24
2 34 3 4 1 34
13 14 23 24
1 12 4 12 3 2
4 14 3 13 13 14 4 24 13 3 23 23
2 3 2 4 1 2 1 3 1 4
14 13 34 24 23
1 4 1 3 3 4 2 4 2 3
Figure 1. The proper part of the face poset of Q(M )
2,1,1
The posets P coincide with the posets obtained from preposets corre-
σ
sponding to normal cones of the matroid base polytope Q(M) studied by
Postnikov, Reiner, and Williams [14].
Billera, Jia, and Reiner [5] define posets related to bases of matroids: For
each e ∈ B ∈ B(M) the basic bond for e in B is the set of e′ ∈ [n]\ B for
which (B \ {e}) ∪ {e′} is another base of M. Dually, for each e′ ∈ [n] \ B
the basic circuit for e in B is the set of e ∈ B for which (B \{e})∪{e′} is
another base of M. Since e′ lies in the basic bond for e if and only if e lies
in the basic circuit of e′, one can define the poset P to be the poset defined
B
on [n] where e < e′ if and only if e′ is in the basic bond for e. The next
PB
proposition shows that our poset P is the same as the poset P of Billera,
σ B
Jia, and Reiner if σ is a vertex e .
B
Proposition 2.11. If B is a base of a matroid M on [n] and σ is a vertex
e of Q(M), then P = P .
B σ B
Proof. Since all connected components of M are singletons, P and P have
σ σ B
the same elements. If e ,e ∈ B, then they are not comparable in P since
1 2 σ
e ⊆ H and e ⊆ H . Also, e ,e ∈/ B are not comparable in P because
B e1 B e2 1 2 σ
9
e ⊆ H and e ⊆ H . Now assume e ∈ B and e ∈/ B. We claim
B B∪{e1} B B∪{e2} 1 2
that e < e if and only if B′ := (B\{e })∪{e } ∈ B(M), i.e., e < e .
1 Pσ 2 1 2 1 PB 2
Assume e < e , i.e., e ∈ S and σ ⊆ H implies e ∈ S. Suppose B′ is not
1 Pσ 2 2 S 1
a base of M. Then r(B′) = r(M)−1. Thus we have e2 ∈ B′,σ ⊆ HB′, but
e ∈/ B′, which is a contradiction. Therefore B′ is a base of M. Conversely,
1
assume B′ ∈ B(M). Let S be a subset of [n] such that e ∈ S and σ ⊆ H ,
2 S
i.e., |B∩S| = r(S). If e ∈/ S, then |B′∩S| = |B∩S|+1 = r(S)+1, which
1
is impossible. Thus e < e . (cid:3)
1 Pσ 2
3. The cd-index
In this section, we define the cd-index for Eulerian posets and give the
relationship among cd-indices of polytopes when a polytope is split by a
hyperplane.
Let P be a graded poset of rank n + 1 with the rank function ρ. For a
subset S of [n], define f (S) to be the number of chains of P whose ranks
P
are exactly given by the set S. The function f : 2[n] → N is called the flag
P
f-vector of P. The flag h-vector of P is defined by the identity
h (S) = (−1)|S\T| ·f (T).
P P
T⊆S
X
Since this identity is equivalent to the relation
f (S) = h (T),
P P
T⊆S
X
the flag f-vector and the flag h-vector contain the same information.
For S ⊆ [n], define the noncommutative ab-monomial u = u ...u ,
S 1 n
where
a if i ∈/ S,
u =
i b if i ∈ S.
(cid:26)
The ab-index of the poset P is defined to be the sum
Ψ(P) = h (S)·u .
P S
S⊆[n]
X
An alternative way of defining the ab-index is as follows. For a chain
c := {ˆ0 < x < ··· < x < ˆ1},
1 k
we give a weight w (c) = w(c) = z ···z , where
P 1 n
b if i ∈ {ρ(x ),...,ρ(x )},
z = 1 k
i a−b otherwise.
(cid:26)
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