*
This research was supported in part by the Air
Research under Contract AFOSR-68-1415.
For~
Office of Scientific
A q-ANALOG OF TH: PARTITION LATTICE
by
T. A. Dowling
Department of Statistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 779
November, 1971
A q-ANALoo
OF THE PARTITION l..AlTICE
T. A. Dowling
Univerosity of Noroth Ca:r>oUna at Chapel Hilt
1.
INTRODUCTION
The set of all partitions of a finite set, when ordered by refinement, is
a well-known geometric lattice enjoying a number of structural properties.
Every upper interval of a partition lattice is a partition lattice, and every
interval is a direct product of partition lattices.
The partitions with a
single non-trivial block form a Boolean sublattice of modular elements, and
the Whitney numbers are the familiar Stirling numbers.
Because of these and
other structural properties, the partition lattices occupy a middle ground between the highly structured modular geometric lattices (projective geometries),
and arbitrary geometric lattices (combinatorial geometries), exhibiting some of
the consequences of the departure from modularity while still retaining enough
of the structure to facilitate their study.
We describe in this article for any prime power
lattices, here called
q-partition lattices,
erties of partition lattices.
q
a class of geometric
which share a number of the prop-
There is a natural order- and rank- preserving
map from the q-partition lattice to the partition lattice of the same rank,
which reduces to an isomorphism when
q. 2.
We examine the interval struc-
ture of the q-partition lattice and obtain a representation of it as the lattice
*
The roesea:r>ch roeporoted he roe tJas parotial'ty supporoted by the Ail' Foroce Office
of Scientific Researoch undero Controact AFOSR-68-1415.
2
of a
subgeometry of a projective geometry over the q-element field.
The
characteristic polynomial and MBbius function are obtained, and a Stir1inglike identity and recursion derived for the Whitney numbers of the q-partition
lattice.
We conclude with an application to the enumeration of factorial de-
signs using the geometrical formulation of the design problem developed by
Professor Bose in his classical 1947 paper [1].
2.
PRELIMINARIES
We summarize in this section a number of results and definitions needed
later [4,7].
An (partially) o~ered
set
(P, s)
is a set
anti-symmetric, transitive relation on
P,
1ation is implicit, we write simply
for
is finite if
The direat
P
is a finite set.
p~duat
(u,v) S (x,y)
iff
an ordered set
P
relation of
aovers
x,y.
x
of two ordered sets
u s x
A subset
if
P
C
Y> x
and
is a ma:x:imaZ ahain in
n
and
[x,y]
(P, s).
When the order re-
An ordered set
(P,S)
v s y
in Q.
with order
pxQ
An intervaZ
[x,y]. {z€plxszsy}
of
(with the order
An element y
such that
[x,y]
consists only of the two elements
of
is a ahain if it is totaZZy ordered:
P
C
is
n,
x S y.
[x,y]
x,y€p
The Zength of
c
one less than its cardinality.
covers
iff
P satisfies the ahain condition if all maximal chains in any inter-
i€[l,n].
val
[x,y]
all
x € P),
denoted
P
=- {xi Ii€[O,n]}
< ••• < x •
written x S y.
P, Q is the set
is the ordered subset
X
o < xl
in
together with a reflexive,
All ordered sets considered here are finite.
P), defined for all
in
P
P
are of the same length.
r(x),
If
P has a zePO eZement
0
(0
S
and satisfies the chain condition, the rank of an element
is the length of a maximal chain in
[O,x].
If
x
for
X€p,
P has a unit
element
of
x~P
(l~x
1
P have a
0
minimal upper bound
y
1.
An atom (aoatom)
~(x) ~ ~(y).
implies
is rank-p!'ese!'ving if
P ~ Q,
Le.
r(Hx»'" rex)
If
ordered sets, then
,
If both
have a unique
~: P~
~
is a lattice and
P
and
~
Q are isomo:rrphia,
~: P~ such that $, ~-l are both
~: P~
~(xvy)
...
is a Zattiae isomo:rrphism.
~: P~
P, Q is a bijection
ordered sets
x~P.
for all
Q is a lattice and
P
if order-preserving iff
P, Q satisfy the chain condition,
iff there is a bijection
order-preserving.
~ (x)A~ (y)
x, y
0
called their join, and a unique maximal lower
xVy,
P, Q are ordered sets, a function
written
is an element covering
called their meet.
xAy,
If
~
and
[x,ll.
P is a Zattiae iff any two elements
(covered by 1).
x
and satisfies the chain condition, the aorank
is the length of a maximal chain in
Let
bound
x~X),
for all
is an isomorphism of
~(x)v~(y),
~(xAy)'"
An anti-isomo:rrphism of two
such that both
~,~
-1
are order-
inverting.
A (finite) lattice
some atom p
of
L.
L is geomet!'ia when
y
covers
iff
y'" xvp
for
A geometric lattice satisfies the chain condition, and
its rank function satisfies the semimodular inequality:
r(x)+r(y).
x
Elements of rank
1, 2, r(1)-2, r(l)-l,
r(xvy)+r(xAy)'"
are called points, Zines,
aoZines, aopoints, respectively.
A (finite) aombinato!'iaZ
geomet~
with a closure operator A.-+ A on
a ~
A
ments of
S,
called points, are closed.
such that
ASS,
B-b;
B
b~B.
for all
called bases of
A basis of
b ~ Aua,
A,
is a finite set
S
together
S satisfying the exahange property:
a ~ AUb,
implies
G'" G(S)
and
such that the empty set and all ele-
An independent set is a set
BS S
All maximal independent subsets of any set
have the same cardinality, called the rank of
G is a basis of
S,
and the rank of
G is the rank of
S.
A.
A
4
subgeomet~
A set
AST
If
H of
G is a subset
is independent in
T of
A ~ AnT.
S with closure operator
H iff it is independent in
G.
G(S) is a combinatorial geometry, the set of closed sets of
by inclusion, is a geometric lattice L(G)
G, ordered
whose points are the elements of
S.
Conversely, every geometric lattice
on its set
G(S)
such that
A + A = {p€SI p $\/ A ail. If H = H(T)
aie:
the lattice L(H) consists of all elements
S of points by
is a sub geometry of
x€L(G)
L defines a combinatorial geometry
G = G(S),
x=\I
a
aie:A i
for some subset
AST.
THE LATTICE OF PARTITIONS
3D
A partition of a finite set
X with
n
elements is a set
of disjoint, nonempty subsets of
The subsets
are the bZoaks of
Xi
n.
X,
such that
There is an obvious correspondence be-
tween partitions of X and equivalence relations defined on
X,
wherein the
blocks of the partition are the equivalence classes.
The set
n
$
IT
n
of all partitions of
X may be ordered by
iff each a-block is a union of n-blocks.
a
With this order,
lattice, with zero element the partition consisting of
corresponding to the identity relation on
1 consisting of the single block
X,
X,
~efinement:
n
IT
n
is a
singleton blocks,
and unit element the partition
corresponding to the universal relation
on X.
The join and meet in
T
= {Yjlje:[l,l)}
IT
n
may be easily found from the inte~seation g~aph I(n,a),
bipartite graph with vertices
iff
xinY
j
of two partitions
is nonempty.
Xi' Y
j
A block of
(ie:[l,k), j€[l,l)
and edges
is an intersection xinY j ,
a
5
edge XiY ,
j
is a union
and a block of
neeted component of
over all Xi
in a con-
l(n,a).
The partition lattice is a geometric lattice, with rank function
n-k(n),
where
ken)
is the number of blocks of
4.
field
F. GF(q) ,
pendent set
spanned by
B{a)
the
THE q-PARTITION LATTICE Qn
and let
B in
B,
a unique subset
B(a)
is the minimal subset of
of
B,
defined by
B whose closure contains
a.
We call
B{a)
a€B.
1 be coordinatized over F,
nby a non-zero list
i. e. each point
lists is linearly independent in
ciating with every point
a€S
n
F •
a€S
is represented
so that a subset
S is independent iff the corresponding set
its coordinate list.
over a finite
l' Given any indenwe may associate with each point a€B, the subspace
S,
B-support of a,
n-l
S be the set of points of IF
Let lP
of
n.
be a projective geometry of dimension
Let lPn-l (F)
r(n)·
{~Ii€[l,k]}
of coordinate
We follow the usual convention of asso-
the set of all non-zero scalar multiples of
Thus every list
~€Fn_{O}
represents a point of
S,
with two lists representing the same point iff they are scalar multipler of
each other
Then if
B
= {bili€[l,k]}
set of coordinate lists of the
for some
a.
Ai€S,
idl,k] ,
ex
II:
is an independent set, with
b ,
i
L~.l
{~Iidl,k]} a
B is the set of all points
Ai ~,
where
ex
a€S
such that
is a coordinate list of
We shall find it convenient to denote this relationship by writing
6
a = L~=l Ai b i ,
with the understanding implicit in all such expressions that
a set of coordinate lists of the
is fixed, so that the
bi
determined up to a constant scalar multiple.
k
a' = r i=l A't b i
A'i = KA
a =
implies
for all
i
L~=l Aib i
a
= a'
iE[l,k].
is a point of
iff there is a
nonempty subset of
Ki
= Ai'
and
biEB l ,
Let
B(a) ,
X ... {xilid1,n]}
B,
{biE:BIAi~O}. We shall frequently
and
Ai
B(a)·
r~=l Aib i
are non zero.
= 0,
of X,
and
be a basis of lP
Thus if
is the point
LKib i ,
B1
is a
where
We define a q-partition of X
n-1"
A= {ailiE[l,k]}
ai'
then no two
is a q-parti tion
~
have non zero
Equivalently, the matrix with rows
elements in the same position.
kxn
LAib i (brEB(a»
A of points for which the X-supports
is a coordinate list of
~
as
Then
bi~Bl·
are disjoint sets.
aEA,
KEF*.
More generally, if
LAib i (b i EB 1 )
(briefly, a q-partition) as a set
X(a),
Ai b i ,
KE:F* = F-O such that
a = Ka,
then
"i
r~"'l
a •
In particular,
find it convenient to write the expression
so that all the coefficients
Thus
are uniquely
Ai
ooZ:umn-monomiaZ TrKltn::c over F,
with no zero rows.
Two
kxn
~
is a
column-
monomial matrices represent the same q-partition A iff one is obtainable
from the other by a permutation of rows and multiplication of rows by non zero
scalars, i.e. by premultiplication by a
It is clear that every q-partition
kxk monomiaZ matrix.
A of
that any subset of a q-partition is one also.
We denote by
partition, as is the empty set.
of X.
(1)
For
X(A)
AEQ,
let
n
=
Y X(a),
k(A)
X is an independent set, and
In particular,
Q
n
the set of all q-partitions
be the cardinality of
XO(A)
•
X is a q-
A,
and let
X - X(A).
aEA
Then.
XO(A)
{X(a)'t aEA}
is a partition of
may be empty.
X(A)
into
k(A)
blocks.
We may remedy this by adjoining to
XO(A)
However,
the empty
?
subspace
(2)
z
'!r(A)
of
Xuz
(or any element not in
{X(a)lacA}
==
into
k(A)+l
Proposi tion 1.
U
x).
Then
AcQ
yields a partition
n
{XO(A)uz}
blocks.
be the map A i+ \I a cA a i from the set of
n
i
q-partitions of X to the lattice of subspaces of lP 1. Then the relation
n(3)
~
A
Let
0: Q +L
n
B iff o(A) s o(B)
defines an order on .Q •
n
rank function
So ordered,
rCA) = n-k(A).
The map
jective and preserves order and rank.
Proof:
Suppose
a == LAib i
X (A) uz
o
(bicB(a»,
of
'Ir(A)
contained in no
'Ir(A) = 'Ir(B) ,
o(A) S o(B).
hence
satisfies the chain condition with
Qn
'Ir: Qn+ITn+l'
If
Then
q
= 2,
A£B,
defined by (2), is sur'If
is an isomorphism.
so every
(bi€B(a».
X(a) == WX(b i )
must then be the union of
XO(B)uz
so
a€A
The remaining block
and the blocks
o(A) = o(B)
X(a).
Thus
'Ir(B) S 'Ir(A) ,
and each
B(a)
is a singleton, i.e.
can be written
A == B.
X(b i )
implies
0
is therefore
injective, so the relation defined by (3) is anti-symmetric, hence an order on
Q.
n
It is evident from the above that
from
Q
n
A> B in
Q
n
iff
A can be obtained
by a sequence of single point deletions and/or replacement of two
bl , bl
of BI (BSB'<A) by a third point b'i+Ablj on the line
j
i
joining b l i and b'j. Each such operation decreases cardinality by one, so
points
the length of all maximal chains in
[B,A]
is
the chain condition and has rank function
'Ir(A)
so
let
has
'Ir
k(A)+l
blocks, the rank of
for each
a i = LXj (xjcXi ),
A= {a Iic [l,k]},
i
so
'Ir(A)
'Ir
idl,k] •
is surjective.
Thus
Q
n
rCA) = k(X)-k(A) = r-k(A).
is
Given any partition
preserves rank.
k(B)-k(A).
T ==
Since
(n+l)-(k(A)+l) == n-k(A) ,
{XOuz ,Xl' ••• ,Xk } of
Then 'Ir(A) ==
If
satisfies
q == 2,
T,
Xuz,
where
A is the only preimage
8
of
hence
T,
n
is a bijection.
Clearly
n
-1
is order-preserving.
o
Remark: One could define a G-partition of a basis X of a combinatorial
geometry
G in the analogous way.
It can be shown that
cr
(3) defines an order, while the chain condition holds iff
is injective, so
G has no trivial
lines.
Corollary 1. A covers B in Qn iff
A
=
B - bi
or
Corollary 2. Each element of rank n-k in Qn is covered by
(~) + (q-l)(~)
elements of rank
n-k+l.
Remark: The partial order defined by (3) can easily be described in
terms of representative matrices of q-partitions.
tative matrices of
A, B,
respectively, then
A
M,
If
~
A
~
are represen-
B iff there is a
MA = P~~. An equivalent definition, independent of dimension, can be obtained by adding n-k(A) zero rows
k(B)xk(A)
column-monomial matrix
P such that
in arbitrary positions to any representative matrix of each element
The
nxn
column-monomial matrices over
F
A
~
so
~
B if SnNA S SnNB.
A
B iff there is a
P€Sn
n
such that
Two matrices
n
form a semigroup Sunder multi-
Mn of monomial matrices.
plication, containing as a maximal subgroup the group
Then
A€Q •
NA
= PNB,
Nl ,N 2€Sn
i.e. iff NA€SnNB'
generate the same
9
principal left ideal of Sn
iff there is an
MEM
such that N = MN " But
2
I
n
represent the same q-partition,
N , N
2
I
is anti-isomor,phic to the set of principal left ideals of S ,
n
this is precisely the requirement that
so
Q
n
ordered by inclusion"
Our next two propositions concern the interval structure of
Qn "
Proposition 2. If BEQn is of cardinality k,
~
then
[B,l]
Qk"
A S B imply the subsets
Proof: In Qn' A ~ B iff A S B. But AEQ,
n
B(a),
aEA,
are disjoint, so
q-partition of
A is a q-partition of
B is a q-partition of
X in
B.
[B,l].
Conversely, every
The closure in
B is
that of IP
n
l' so this correspondence preserves order, and the subgeometry on
nis isomorphic to ]Pk-l.
o
IXO(A) I
Proposition 3. Let A· {atltE[l,k]}EQn' and let
= nl
Ix(at)l
,
lE[l,k].
Then
[O,A]
x
Proof: Let Xo = XO(A), Xl
al = LKix i
where
tE[l,k] ,
XOUZ,
Namely, if
(XtEX ).
lm
Xtm
blocks
order in
B ,
t
lE[l,k],
UBl
X
t
x II
•
Suppose
ldl,k] •
n(B) :;; n(A)
~
and
But given only the
alEBl
n(B)-blocks not in
are uniquely determined by the
(lE[l,k]),
for all
then
him
al.
= LKix i
as determined by the blocks of
n(B)
an arbitrary q-partition
The order in
Xl'
B :;; A iff
•••
are the n(B)-blocks contained in Xl'
To the set
not in XOuz,
X(al)'
=
B = {bEBIX(b)SX }·
l
l
the subsets
BE[O,A].
Then
(xiEXt )·
= nO'
lE[l,k],
[O,A]
B of X
o can be added to obtain a
O
is clearly the product of the orders in the
and the set
is that of
X •
O
It is evident from the above that the
while that of
X
o is
o
10
Corollary 1. Let B ~ A in Qn. where A = {at1t€[1.k]}. Let
IBO(A) I
..
mO.
IB(at) I
= mt.
tdl.k].
Then
x
Corollary 2. If at" LKixi
[O.A]
(xi€xt ),
••• x IT
~
•
A = {at1t€[1.k]},
the atoms of
are
Co ro 11 ary 3. Let A" {a} be a copoint of Q.
If X(a)" X,
n
[O.A] ~ nn.
while if
X(a)" {xi}.
We prove in §5 that
F
n-
1.
Q
n
[O.A] ~ Qn-l.
is isomorphic to the lattice of a subgeometry of
The next proposition (and its corollary) is not required for that proof.
but we include it to describe the nature of joins and meets in Q.
n
Proposition 4. Qn is a lattice.
Proof. Let A.B€Q.
Since
n
~
preserves order.
~(C) ~ ~(A)VTI(B)
for any upper bound C and lower bound
TI(A)VTI(B)
= {Xtlt€[l,m]}u{xOuz}.
the blocks of
X-support
TI (A)VTI (B)
X • where
t
Suppose
{Xtlt€[l,p]}
At ,B
t
are the subsets of A,B.
D of A,B€Qn • Let
where
for which there exists a point
and
p ~ m.
are
Ct€AtnBt with
respectively. with
X(At ) = Xt. X(Bt )" Xt· Then clearly c" {ct1t€[l,p]} is a minimal upper
bound of A,B in Q. To show that A v B exists. we must prove that the
n
c
t
are uniquely defined.
For a fixed
t€[l.p].
let
At" {aili€[l.r]}.
11
B - {bjlj€[l,s]}.
t
X-support
Xt.
c'
r
=
t
-
L Ki a i
i=l
r
,
L
K iai
i=l
= xt ,
s
L
A bj ,
j
L
A'jb j ,
j-l
s
-=
j-=l
say, where all coefficients are non-zero.
x
with
Then
Ct ...
say
ct,C't in AtnBt
Suppose there are two points
Let
a i '" LaiuX u '
b j - LSjvXv.
If
is contained in the X-supports of
.
...
respectively.
Thus
x(a ) n X(b )
i
j
is nonempty.
Ki/A j
Since
section graph of the
X(a ),
i
Thus all the ratios
K'i/Ki'
The blocks of
'II'(A)
A'j/A j
'II'(A)ATT(B)
such that
If
Yt = XO(A)nX(b),
'II'(B).
Y = X(a)nXO(B),
t
Y ... X(a)nX(b),
t
valence relation
let
z~Yt.
let
is a block of
and the
a'" LKiXi
A,B
whenever
'II'(A)VTT(B),
the inter-
j€[l,s]
is connected.
X(b j)'
Ct'" C't and AVB = C.
(xi€X(a»,
We define a q-partition
D contain the point
iff
(xi€Y.e.m).
b - LAix i
respectively, and let
D contain the point
XiEX j
K' /K ... A'j/A
j
i i
are equal, so
Let
then partition Y
t
E:
so
are the nonempty intersections of the blocks of
be arbitrary points of
of
that
x
t
i€[l,r],
TT(A)A'II'(B)
with the blocks of
(xi€X(b»
K'i/A'j'
LAiX
LKiXi
into the blocks
Y
t
D of
(xi€Y ),
t
i
(xi€Yt ).
Y
tm
be a block
X as follows.
while if
Finally, if
defined by the equi-
Ai/K
... Aj/K • For each Y
let D conj
tm
i
It is clear (see the proof of Proposition 3)
D is the unique maximal upper bound of
A,B
in
Q,
n
i.e.
D = AAB.
o
Corollary.
Qn
is a geometric lattice.
12
Proof:
By Corollary 1 of Proposition 1,
or A = B-{b ,b }u{b +Ab}.
r s
r
s
XiEX(b r ),
A covers
In the first case,
while in the second case
A
A
D
B iff
BV(X-x )
i
A D B-b ,
r
for any
= BV(X-{xi,xj}U{Xi+AXj })
for any
XiEX(b r ), xjEX(b s )'
o
Sa
In the lattice
L
n
of subspaces of F n _ l , let X*
set of copoints defined by
every copoint
point, then
copoint
c*
a :!> c
= LAixi •
iff
L
~L
Since
A
n
~
= LK i x* i
at~ a*,
is the dual basis of Xi
such that a= LAiXi
a
c = LAiX*i
c ~ c*
be the
= LKix i
is a
is the
is the point
extend to an antiiso-
•
n n'
0:
Qn~Ln
defined by
o(A)
= \/ai
The ele-
(aiEA).
are independent sets, so
B in
isomorphic to
c
and the dual of a copoint
Recall the injection
Q
X*
(XjEX-X i )·
= {x*i1iE[l,n]}
The dual of a point
LKiA i = O.
The pair of bijections
*:
ments of
x* i =Vx j
may be written as
= LKiX*i'
a*
morphism
eEL
REPRESENTATION OF Qn
Q
n
iff
o(A):!> o(B),
the image of
Q
n
under 0
Qn • Thus the image Rn of the composite map 0*
is isomorphic to
Qn • Further,
=
n - r (0 (A»
L
=
is anti-
= *00:
Q ~L
n n
13
so the rank of an element in
R
n
is its rank in
We now prove that
R
n
is a geometric lattice, isomorphic to its image
Rn
is the lattice of a subgeometry of lP
Proposit;on 5.
Qn
0'*: Q +L •
under the map
n
R
n
n
n-
L.
n
1.
is the lattice of the subgeometry of lPn- 1
on
the point set
consisting of
X and all points
Xi+AX
on the lines joining two points of
j
X.
Proof:
We have only to show that
and that a subspace
The O'*-image of A
{brlre[l,n-k]}
re[l,n-k],
is
(4)
iff it contains a basis in
n
Vb
is
X-Xi
(rE:[l,n-k])
r
b
r
~ a*,e
l
~
Let
for all
Consider now the image of the atoms of Q.
n
iff
K,e
Xi+AX j
is
x*i+AX*j.
=0
for all
l
~
i.
A point
The dual of
= LK,eX,e
a
is the
Thus
a
= LKlKl
i,j,
and
is the a*-image of
Ki+AK j
= O.
X-{xi,Xj}U{Xi+AX }
j
is the set of points of
P.
n
A = {allle[l,k]}eQn'
where
are
-1
K
i Xi -
-1
K
j
xj
iff
K
l
=0
for
Hence
al = LKiXi
(xieXl
Corollary 1 of Proposition 3, (4), and (5), the points of
(6)
where
a*(X-x i )
The point
all
R
= {a,eI,eE:[l,k] h:Qn
and that of
O'*-image of
is in
is the O'*-image of the atoms of
is any independent set such that
,ee[l,k].
X* i
U
52
= X(al».
[O,a*(A)]
By
in
P
n
14
Fix a point, say
xl'
in
Xl'
for each l€[l,k],
and let
Then T· T UT 2U••• UT UX
is independent and of cardinality n-k· rL(cr*(A»,
1
k O
so
T is a basis of
cr*(A)
in
S2'
It remains to show that for every
(6) and (7).
xi+Axj€U
Let
or
for all
Let
Xi€X-XO'
If
If xi+Axj€U,
xi+Kxj€U, A ~ K,
xi+Ax j ,
Xi,Xj€X-X '
O
in U on the line joining Xi
Define now a directed graph
1
and
Xj€XO'
then
m has
either
Xi,Xj€U,
so
and
Hence every point Xi+Ax j
We conclude that for every pair
xi+Axj
is of the form given by
2
X • {xilxi€U}. Then
O
If X • X, u. 1 - cr*(l), so assume
O
Xi,Xj€X •
O
A€F*,
a contradiction.
Xi,Xj€X-XO'
UnS
U be a subspace, and let
X-XO is nonempty.
Xi€XO'
U€L ,
n
in
then
Xi€U,
so
Xi,Xj€XO
Xi,Xj€XO•
there is at most one point
xj '
D with vertex set
X-X
such that for each
O
labelled A and one edge
point xi+Ax
in U, D has one edge Xi +A x j
j
A- 1
-1
x ~ Xi labelled A • Then any two vertices in D are joined by no edges
j
or exactly two, with opposite orientations, and representing the same point.
Suppose that
is a path of length two in
is an edge
1
This
Xl
idea~
-KA
----~
Xi
D.
in D.
Then
(xl+Axj)-A(Xj+Kxi ) - Xl-AKxi€U,
so there
We conclude that every connected ·component
aZthough simi l,ar to our original,
proof~
is suggested in [9].
D ,
l
15
lE[l,k]
say, is a complete graph.
Xl
Let
Dl • To
can be assigned
be the vertex set of
complete the proof, we need only verify that the points
labels
K-li€F*
xi ~ x
so that
Fix a point, say
xl'
Xl'
in
j
in
D
l
implies
(see (6».
A·
and assign the label 1 to
xl.
To each
-1
Xi€Xl-Xl assign the label, say -K i' of the edge xl -+ xi' Then
-K-1
-K- 1i
-1
-1
-1
A
j;> x
and xl
imply K
so A • -KiK j '
•
-K
iA,
> xi -+ xj
xl
j
j
as required.
o
Remark: With the exception of Corollary 2 of Proposition 1, the assumed
finiteness of
F has not been used.
Hence all our results to this point hold
for an arbitrary field.
6,
THE CHARACTERISTIC POLYNOMIAL AND ~"'UTNEY NUt-BERS OF Qn
A modutap element of a geometric lattice L with rank function
.element
x€L
such that
r(xvy)+r(xAy)
modular element, the map
inverse w * wAy,
z
~
for any
xVz
y€L.
= r(x)+r(y)
for all
is an isomorphism
[xAy,y]
y€L.
~
r
If
is an
x
is a
[x,xVy]
with
Every point of a geometric lattice is a
modular element.
Proposition 6. In the geometric lattice Qn of q-partitions of X,
subset
B(X).
M· {AIA£X}
is a sublattice anti-isomorphic to the Boolean algebra
M is modular in
Every element of
Proof. If A€M, B€Qn ,
{a },
i
a
X(b) n (X-A) ,
b € B,
i
Q •
n
the blocks of 1T(A)A1T(B)
€ A,
(8)
and the blocks of
the
1T(A)V1T(B)
X(b)
~
A,
not containing
z
are
not containing
z
are
16
(9)
X{b),
b
B,
€
X{b) S A.
~(AAB). ~(A)A~{B)
It is clear (see the proof of Proposition 4) that
~(A)V~(B)
so
= ~(A)v~(B).
k(A)+k(B)
also
B€M,
The total number of blocks in (8) and (9) is
= k(AvB)+k(AAB).
then
= {b},
X{b)
A~
sub1attice, and
Thus since
so
r{C)
= AUB€M,
AAB
= n-k(C),
AVB
and
k(A)+k(B),
A is modular.
= AnB€M.
If
M is a
Thus
is an anti-isomorphism M ~ B(X).
X-A
o
The
funation
~bius
= -Lz:x~z<y
LxL ~ Z
= 1,
~(x,x)
defined recursively by
~(x,y)
~:
~(x,z)
~(x,y)
x ~ y
if
rank n with rank function
of a finite partially ordered set
[7].
=0
If
if x
~
y,
=
l\
~
is
and
L is a geometric lattice of
the aharaateristia poZynomiaZ of L
r,
L
is
(O)
,x v n-r{x) •
x€L
The characteristic polynomial extends to geometric lattices the notion of the
chromatic polynomial of a graph.
In particular, if
tractions [7] of a linear graph
polynomial of
is
G
x(v)
G with k
= vkp(v).
80
pn (v)
of
c
is a copoint of
(lO) vP[O,c](v)
where
IT +
n 1
is the lat-
K + with chromatic polynomial
n 1
IT +
n 1
is
We may obtain the characteristic polynomial
Qn with the aid of the following special case of a theorem of
Crapo ([3], Th.6, Cor.S):
and
components, then the chromatic
the characteristic polynomial of
= (V-1){n)'
(v-1){v-2) ••• {v-n)
is the lattice of con-
The partition lattice
tice of contractions of the complete graph
v{v-1) ••• {v-n),
L
P[a,b]{v)
L,
L
•
If
x:XAC=O
L is a finite geometric lattice of rank n
then
p[x l]{v),
'
is the characteristic polynomial of the interval
Proposition 7.
The characteristic polynomial of
Qn
is
[a,b] of
L.
17
=
(11) p (v)
n
n-l
1T
i=O
Proof:
We take as our
c
Corollary 2, Proposition 3,
B=0
(i.e.
B
= X)
(q-l)
n
v-I
(q-l) (n) •
in (10) the copoint
[O,C]
~
Qn-l.
Since
of Q.
C· {xl}
n
C is modular,
B is an atom of Q not in [O,C].
n
l+(q-l) (n-l). By Proposition 2, [B,l]~ Qn-l'
such atoms is
Be:Q.
=
(v-l-(q-l)i)
or
By
BAC = 0
iff
The number of
for every atom
Thus
n
=
(12) Pn(v)
Since
Pl(v}
Remark:
(v-l-(q-l)(n-l»
= v-I,
p l(v).
n-
o
we obtain (11).
Stanley [10,11] has recently investigated the class of geometric
xl < ••• < xn = 1 of modular
Such lattices, called supersoZvabZe lattices, have the property that
lattices containing a maximal chain 0
elements.
= Xo <
all zeros of the characteristic polynomial are positive integers, namely,
where
6,
a
Q
n
i is the number of atoms of [O,Xi ] not in
is supersolvable, with a = l+(q-l)(i-l).
i
Corollary 1. Let
~
n
= ~(O,l).
[O,x _ ].
i l
be the Mobius function of Q ,
~
n
By Proposition
and let
Then
n-l
~
n
=
=
where
= x(x+l) ••• (x+k-l).
x(k)
Proof:
When
1T (l+(q-l)1)
i=O
(_(q_l»n (l/(q-l»(n),
(_l)n
Set
q
= 2,
v· 0
in (11).
~n· (_l)nnl
= ~'n+l
(say),
where
~'n+l
= ~(O,l)
for the
18
partition lattice
IT + •
n l
Since the Mobius function is multiplicative on direct
products, we obtain from Corollary 1 of Proposition 3,
Corollary 2. Let B
B ... {bjlj€[l,m]}.
S A
in Qn'
m = IBO(A)I,
O
Let
where
m
i
A = {aili€[l,k]},
= IBO(ai)l,
i€[l,k].
Then
jJ (B ,A)
The Whitney numbers of a finite geometric lattice L of rank
n
are de-
fined by
(13) w(n,k)
=
L
jJ(O,x)
(First kind),
x:r(x)"'n..,k
the coefficient of
(14) W(n,k)
=
v
k
in the characteristic polynomial, and
L
(Second kind),
1,
x:r(x)"'n-k
the number of elements of corank
If
L'" B,
n
=
L ... L(Vn ) -~ L(lP n- 1)'
space over
GF(q) ,
n
(k)q
(15)
Finally, if
(_l)n-k (~),
W(n,k)
...
the lattice of subspaces of a vector (projective)
then
w(n,k)
where
Some classical examples are the following.
the lattice of subsets of an n-set,
w(n,k)
If
k.
W(n,k)
=
...
are the Gaussian coefficients [6],
(qn_l)(qn-l_l) ••• (qn-k+l_ l )
k
k-l
(q -l)(q
-l) ••• (q-l)
L'" IT + ,
n l
the lattice of partitions of an n-set,
19
=
w(n,k)
s(n+l,k+l),
W(n,k)
=
S(n+l,k+l),
the Stirling numbers of the first and second kind, respectively.
as well as the q-partition lattices
Qn'
are classes of lattices which satisfy
the hypotheses of the following proposition.
as
a = b
or
a
~
All of these,
Here
o(a,b)
=1
or
°
according
b.
Proposition 8. Let {pn In=O,l, ••• } be a class of finite geometric lattices with the property that
X€p
of corank
n
numbers of
(16)
P,
n
L W(n,k)
k,
P
n
k€[O,n],
k€[O,n].
is of rank n
n€[O,oo).
Let
and
Pk for all
W(n,k) be the Whitney
[x,l]
w(n,k),
~
Then
w(k,m)
= o(n,m),
W(k,m)
= o(n,m).
k
and
(17)
L w(n,k)
k
The numbers
(18) a
n
=
W(n,k), w(n,k)
L W(n,k)b k ,
k
then satisfy the inverse relations
b
n
=
L w(n,k)a1<. •
k
= L ..
Proof: We use the identities o(O,y)
x~y
Then
L W(n,k)
w(k,m)
=
k
L
X€p
L
n
= L
y€p
=
O(m,n-r(y»
o(m,n-r(y»
n
= o(n,m).
Similarly,
L
x:x~y
n
L
y€p
ll(X,y) o(m,n-r(y»
y:y~x
o(O,y)
ll(X,y)
ll(O,X).
20
L w(n,k)
W(k,m)
...
l.l (0 ,x)
L
x€p
k
n
= L
o (m,n-r(y»
yEP
...
L w(n,k)a
k
l.l(O,x)
L
x:x~y
n
o (m,n-r(y»
L
yEP
=
15 (m,n-r(y»
L
y:y~x
o(O,y)
n
O(n,m) •
...
K
L w(n,k) L W(k,m)b
m
k
... L b L w(n,k)
m
...
m
W(k,m)
mk
Lbo
m nm
m
...
b •
n
The converse is proved analogously.
o
Corollary. The Whitney numbers T(n,k), t(n,k) of the q-partition lattice
Qn
satisfy the inverse relations
(q_l)n
(V-I)
. . L t(n,k)
q-1) (n)
V
n
...
L T(n,k)
k
(q_l)k
k
Proof: Set
a
n
... v
n
in (18).
k
v ,
Then
b
n
(V-I)
q-l (k)'
is the characteristic poly-
nomia1 (11).
o
Note that on setting
q'" 2
and multiplying both equations above by
we obtain the defining relations of the Stirling numbers.
v,
21
Proposition 9. The numbers T(n,k), t(n,k) satisfy the recursions
(19) T(n,k)
•
T(n-1,k-1) + (l+(q-1)(k-l»
T(n-l,k)
(20) t(n,k)
=
t(n-l,k-l) - (1+(q-1)(n-l»
t(n-l,k).
Proof: Every q-partition of X of size k is obtainable either from a
unique q-partition of
unique q-partition
X-x
B of
(AEF*),
n
of size
X-x
n
k-1
x ,
by adding the point
of size k-l
n
by replacing some
or else is equal to a q-partition of
or from a
biEB
X-x
n
by a
of size
k.
This proves (19), while (20) follows from a comparison of the coefficients of
vk
in (12).
o
7. AN APPLICATION TO DeSIGN
We conclude with an application to a problem of design in statistics.
Consider an experiment in which
q
levels, where
q
n
factors are to be observed, each factor at
is a prime power.
The
qn
different combinations of
levels of the factors are to be partitioned into
qn-r blocks of size
r
q ,
in such a way that no t-factor or lower interaction is confounded with blocks.
As
Bose [1,2] has shown, the problem may be represented geometrically by rep-
resenting the main effects of the
~
n-
lover
GF(q).
n
factors by the points of a basis
Each t-factor interaction is then represented by a point
of X-weight (cardinality of its X-support)
choice of a subspace
of
X of
U of F _
n l
t.
of dimension
The design is specified by the
n-r-1,
in which the points
U represent the interactions confounded with blocks in the design.
the design will confound no interactions of
tains no points of
t
or fewer factors iff
Thus
U con-
22
=
For given
n, t,
it is desirable to maximize the dimension
as to minimize the block size
r
q •
(n,n-r)
linear code over
(C ,C 2 , ••• ,C )
l
r
that the number of lists
no point of a given spanning set
where
p(v)
T of
of projective hyperplanes such that
~n-l
of closed sets in the subgeometry of F
r
p(q )
imum dimension of a subspace of F
is called the critical
(~)q
where
is in every
is
Ci
n-
1
of
N
n-r
iff
on the point set
n-l
r < c,
where
T.
This
is the max-
n-c-l
containing no points of
T.
T.
The integer
We show in [5] by an application of
of
(n-r-l)-dimensional
projective
T is then given by
[~(qr_qi)]
N
i=O
n-r
(21)
=0
~xponent
M5bius inversion that the number
subspaces not meeting
It is shown in [4]
is the characteristic polynomial of the lattice
implies in particular that
c
with minimum
may be considered as a special
case of the critical problem of combinatorial geometry [4].
L(T)
GF(q)
t+l.
The problem of determining the maximum k
(q_l)-rp(qr),
of U so
Bose observed [2] that the problem is
equivalent to that of finding an
distance at least
n-r-l
=
f
i=O
is the Gaussian coefficient (15).
The critical exponent of the
is well-known to be the smallest r such that n ~ (~)q =
2
(qr_l)!(q_l). We may, in addition, obtain the numbers N
in this case,
n-r
set
since
8
Qn
~
L(S2).
We state this result as
Proposition 10. The number Nn-r of
n
factors at
q
levels each, in
qn-r
n
r
(q ,q )
factorial designs (i.e.
blocks of size
main effects or two-factor interactions are ccnfounded is
qr)
such that no
23
r
(_l)i
i=O
r-1
N
n-r
=
(
r ]
s...::l
q-1 (n)
r-1
1f
(qr_qi)
i=O
Note that
N
n-r
=0
r
iff
n > q -11.
q-
For the critical expoennt
c,
we
have the
Coroll ary.
The number of
n
c
(q ,q )
designs with minimum block size
c
q ,
such that no main effects or two factor interactions are confounded, is
(q-1) n
(q c-1]
q-1 (n)
Nn-c
=
REFERENCES
Design~"
[1]
Bose, R.C., "Mathematical Theory of the Symmetrical Factorial
Sankhya, 8 (1947), 107-166.
[2]
Bose, R.C., "On Some Connections between the Design of Experiments and
Information Theory." BuZZ. Int. Stat. Inst., 38 (1961), 257-271.
[3]
Crapo, H., "l1obius Inversion in lattices." Arah. derMath.
(1968), 595-607.
[4]
Crapo, H. and G.-C. Rota, Corribinatorial Geometries, MIT Press,
Cambridge, Mass., 1970.
[5]
Dowling, T.A., "Codes, Packings and the Critical Problem," in Proaeedings of the Conferenae on Combinatorial Geometry and Its AppZiaations (A. Bar1otti, ed.), Perugia, Italy, 1971.
[6]
Goldman, ..T. and G.-C. Rota, "On the Foundations of Combinatorial Theory
IV: Finite Vector Spaces and Eulerian Generating Functions." Sutd.
Appl. Math. XLIX (1970), 239-258.
[7]
Rota, G. -C. , "On the Foundations of Combinatorial Theory I: Theory of
Mobius Functions." Z. f{ahrsaheinZiahkeitstheorie und venv. Gebiete,
2 (1964), 340-368.
XIX
24
·e
[8]
Rota, G.-C., "The Number of Partitions of a Set,"
71 (1964), 498-504.
[9]
Rota, G.-C., P. Doubilet and R. Stanley,
Function," (to appear).
[10]
Stanley, R.,
"Modular Elements in Geometric Lattices," (to appear).
[11]
Stanley, R.,
"Supersolvable Semimodular Lattices,"
Am. Math. MonthZy
"The Idea of Generating
(to appear).