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F-Binomial Coefficients and Related Combinatorial Topics: Perfect Matroid Designs, Posets of Full Binomial Type and F-Geodetic Graphs
Annals of Discrete Mathematics 30 (1986) 143-1 58 0 Elsevier Science Publishers B.V. (North-Holland)
143
F-BINOMIAL COEFFICIENTS AND RELATED COMBINATORIAL TOPICS: PERFECT MATROID DESIGNS, POSETS OF FULL BINOMIAL TYPE AND F-GEODETIC GRAPHS P i e r V i t t o r i o C e c c h e r i n i and Anna Sappa D i p a r t i m e n t o d i Matematica "G. Castelnuovo" Uni v e r s i t l d i Roma "La Sapi enza" C i t t a U n i v e r s i t a r i a , 00100 Roma, I t a l y
We i n t r o d u c e F-binomial c o e f f i c i e n t s as a n a t u r a l g e n e r a l i z a t i o n o f b i n o m i a l and q - b i n o m i a l c o e f f i c i e n t s . A g e n e r a l c a l c u l u s w i t h these numbers l e a d s t o u n i f y t h e a r i t h m e t i c a l p r o p e r t i e s o f ( f i n i t e ) p r o j e c t i v e and a f f i n e spaces and o f S t e i n e r systems S(t,k,v) i n t o those o f p e r f e c t matroid designs ( 5 2 ) . P a r t i a l l y o r d e r e d s e t o f f u l l b i n o m i a l t y p e ( 5 3 ) and graphs such t h a t t h e number o f geodesics between any two v e r t i c e s depends o n l y on t h e i r d i s t a n c e ( 5 4 ) a r e a l s o s t u d i e d by means o f t h i s f o r m a l c a l c u l u s .
1. F-BINOMIAL COEFFICIENTS Let N (resp.
Q ) denote t h e s e t o f non n e g a t i v e i n t e g e r s ( r e s p . r a t i o n a l
numbers) and l e t be N* = N \ ( O I , L e t F: N
+
Q*
=
Q\rO).
Q* be any f u n c t i o n such t h a t F ( 0 )
any f u n c t i o n such t h a t f ( 0 ) = 0, f ( 1 )
=
1 and f ( N * )
=
F ( 1 ) = 1 and l e t f : N
+
Q be
5 Q*.
F and f w i l l be f u n c t i o n s s a t i s f y i n g t h e above c o n d i t i o n s .
I n what f o l l o w s ,
Given an F, t h e a s s o c i a t e d f w i l l be d e f i n e d by: f(0)
0, f ( n )
F(n)/F(n-l) for n 1 1 .
=
Given an f, t h e a s s o c i a t e d F w i l l be d e f i n e d by: F ( 0 ) = 1, F ( n ) = f ( n ) F ( n - 1 )
i . e . F(n) = f ( n ) f ( n - l ) . . , f ( l )
Two such f u n c t i o n s F and f a r e t h e n m u t u a l l y associated,
f o r n 31.
and sometimes below t h e y
w i l l be used i n t e r c h a n g e a b l y . DEFINITION 1.1.
Given a p a i r ( F , f ) o f m u t u a l l y a s s o c i a t e d f u n c t i o n s and g i v e n
any i n t e g e r s k,n w i t h
OC
n
k < n , we s h a l l d e f i n e t h e F-binomial c o e f f i c i e n t ( o r
b i n o m i a l c o e f f i c i e n t ) I 1 as t h e r a t i o n a l number k F
f-
144
P. V. Ceccherini and A. Sappa
These numbers t u r n o u t t o be p o s i t i v e i n t e g e r s i n t h e f o l l o w i n g examples ( i n which f and F a l s o t a k e i n t e g e r v a l u e s ) .
EXAMPLE 1.2.
Binomial c o e f f i c i e n t s . Consider t h e p a i r o f m u t u a l l y a s s o c i -
ated functions
fl(t)
=
and
t
F ( t ) = t! 1
for a l l tEN.
Then n
in)= k F1
= Ik} i s t h e usual b i n o m i a l c o e f f i c i e n t
k fl
(O
EXAMPLE 1.3. Gaussian numbers. Given an i n t e g e r q > 1 , c o n s i d e r t h e p a i r o f m u t u a l l y associated f u n c t i o n s
f (t) = [ t l 9 9 where [ ]
9
[Ol,!
F ( t ) = [t],! 9
and [ ] ! a r e d e f i n e d by q
9
[OI
and
=
[ll
0,
=
=
9
[ll ! q
[tlq
1,
= q
t-1
t
... + q t l ,
[ t l q l= [ t l [ t - 1 1
= 1,
q
q
... [ llq,
t 22.
Then
n - n I k I F - ikIf 9 q
n
[,Iq i s
the q-binomial coefficient
EXAMPLE 1.4. Constant c o e f f i c i e n t s .
( g a u s s i a n number, c f . 141).
Given an i n t e g e r a z 1, c o n s i d e r t h e
p a i r o f mutually associated functions f(0) = 0,
f(1)
=
1,
f ( t ) = a;
F ( 0 ) = F ( 1 ) = 1,
F ( t ) = at - ’
(tz2).
Then n n I 1 = i l =1, n F OF
and
n
IkjF =
a
for
Usual b i n o m i a l i d e n t i t i e s and q-binomial
O < k
are p a r t i c u -
I45
F-Binomial Coefficients and Related Combinatorial Topics l a r cases o f F-binomial i d e n t i t i e s ( o b t a i n e d f o r F
F
=
1 g i v e now some o f them, w r i t t e n a s f - b i n o m i a l i d e n t i t i e s .
and f o r F
=
F r e s p . ) . We q
P R O P O S I T I O N 1.5. The f o l l o w i n g f - b i n o m i a l i d e n t i t i e s h o l d :
n ‘k’f
- f(n) - fo
n
-
Ikff
-
n-1
I k
{
If +
n-1 k If
-
f(n)
fo
f(n)-f(n-k) f(k)
n-1 ‘k-l’f
- f(n-k+l) n f(k) ‘k-l’f
(O< k tn),
n-1 (k-l’f
These f o r m u l a s suggest t h e f o l l o w i n g DEFINITION 1.6. Given a f u n c t i o n f and any i n t e g e r s k,n w i t h 0 < k
PROPOSITIONS 1.7. L e t
equivalent: (a)
dfn,k
(c)
be as Def. 1.6. The f o l l o w i n g c o n d i t i o n s a r e
i s independent o f n;
n,k
(b) f = f
n,k
9
f o r some q E Q * (where f
= qk
f o r some
f (n)-f (k) 9 9 f q(n-k) bf
n,l
f(n) =
=
=
9
n-1
+ . . . +q
..+q+l
qn-k-1,.
i s f o r m a l l y d e f i n e d as i n Ex. 1 . 3 ) .
q E Q*.
P r o o f . O b v i o u s l y ( c ) = + ( a ) . We have ( b ) -
q
f 9
( c ) because
=
gf
n,k
A
n,k
k
(b).
= qk. We prove now t h a t ( a )
=
L e t us p u t
q f o r a l l n, Then f ( 1 ) = 1 and f o r a l l i n t e g e r s n 3 2 we g e t Af
n,l
f(n-l)+f(l),
so t h a t t h e e q u a l i t y f ( n )
=
q
n-1
+
...
+ q+l f o l l o w s by
induction.
2 . PERFECT M A T R O I O DESIGNS A c o m b i n a t o r i a l geometry ( o r s i m p l e m a t r o i d ) M on a f i n i t e s e t S i s a m a t r o i d
P.V. Ceccherini and A. Sappa
146
0, S andevery s i n g l e t o n o f S a r e f l a t s o f M ( c f . [ 2 1 , [911. A p e r f e c t m a t r o i d d e s i g n (PMD) i s a c o m b i n a t o r i a l geometry M such t h a t e v e r y k - f l a t ( f l a t of rank k ) has t h e same number f ( k ) o f p o i n t s , k = O,l, ...,r k M
on S such t h a t
[lo],
(cf. f(0)
[ 1 2 ] , [ 1 4 1 ) . We s h a l l say t h a t f i s t h e s i z e - f u n c t i o n o f M. Obviously
n
0 and f ( 1 ) = 1, so t h a t f - b i n o m i a l c o e f f i c i e n t s t k l f can be
=
considered
(Osk(n4r-k MI. X PMDs i n c l u d e boolean s e t s 2 , p r o j e c t i v e spaces, a f f i n e spaces, t - ( v , k , l )
designs ( c f . [ 1 4 ] ) . PROPOSITION 2.1.
L e t M be a PMD on a f i n i t e s e t S w i t h s i z e - f u n c t i o n f .
I n t h i s case ( a ) M i s t h e m a t r o i d o f a l l subsets o f S i f and o n l y i f f = f 1' n n f f ( n ) = n, 1 ~ = 1( k ) ~ and = 1.
( b ) M i s t h e m a t r o i d o f f l a t s o f a p r o j e c t i v e space o f o r d e r q i f and o n l y n n-1 n i f f = f ( w i t h q a 2 ) . I n t h i s case f ( n ) = q t ... t q t l , 1 ~ = 1[ 1 ~ and f ",k
9
k q
(Conversely, each o f t h e s e e q u a l i t i e s i m p l i e s f = f 1. 9 n-1. ( c ) IfM i s thematroid o f f l a t s o f an a f f i n e space o f o r d e r q, t h e n f ( n ) = q , n f k 2k-n t h e converse i s t r u e when q a 4. I n t h i s case { k ] f = qk(n-kl and = q - q . n-1 1. (Conversely, each o f these e q u a l i t i e s i m p l i e s f ( n ) = q
q
=
*
P r o o f . ( a ) i s obvious. For ( b ) ( r e s p . ( c ) ) , we have t o prove o n l y t h e " i f " p a r t . A s i m p l e c o u n t i n g argument shows t h a t e v e r y 3 - f l a t i s a p r o j e c t i v e ( r e s p . an a f f i n e ) p l a n e o f o r d e r q, so t h a t t h e r e s u l t f o l l o w s f r o m a w e l l known charact e r i z a t i o n o f p r o j e c t i v e ( r e s p . a f f i n e ) spaces by means o f planes, c f . [ Z ] ( r e s p .
[ll).
0
With an argument which i s s t a n d a r d f o r f i n i t e p r o j e c t i v e spaces ( c f . [14] one can prove t h e f o l l o w i n g PROPOSITION 2.2.
L e t M be a PMD w i t h s i z e - f u n c t i o n f.
( a ) The number o f k - f l a t s i n c l u d e d i n a r - f l a t ( w i t h O < k 6 r ) i s g i v e n by
- 'r, 1 " ' ' r , k - l A k, 1 * k, k-1
'
...f ( r - k + l )
f(r)
.
f(k). .f(l)
--
A ~ , ~ . . . A ~ , ~ - ~
'k, 1
"
'Ak,k-l
'k'f'
( b ) When a k - f l a t i s i n c l u d e d i n a r - f l a t , t h e number o f j - f l a t s between them
(01:k < j c r ) i s g i v e n by
147
F-Binomial Coefficients and Related Cornbinatorial Topics
I n p a r t i c u l a r B(O,j,r) B(k,ktl,r)
= cr(j,r)
and
"" f ( r - k ) A
=
k+l, k
( c ) When a k - f l a t A i s i n c l u d e d i n a r - f l a t 6, t h e number o f maximal c h a i n s
s g i v e n by
o f f l a t s between them ;(k,r)
...B ( r - 2 , r - l , r )
r ) B(ktl,kt2,r)
B(k,ktl
=
=
Ar,k.'.Ar,r-2 Ak
t l ,k ' * ' A r -,lr - 2
F(r-k)
where F i s a s s o c i a t e d t o f . L e t M be a PMD on a f i n i t e s e t S w i t h s i z e - f u n c t i o n f (and
PROPOSITIONS 2.3.
a s s o c i a t e d f u n c t i o n F ) and l e t a ( k , r ) , t i o n 2.2.
B(k,j,r),
;(k,r)
be d e f i n e d as i n p r o p o s-i
The f o l l o w i n g c o n d i t i o n s a r e e q u i v a l e n t :
( 1 ) M i s t h e PMD o f a g r a p h i c space S o f o r d e r q ( i . e . a p r o j e c t i v e space o f o r d e r q ( p o s s i b l y a " l i n e " ) , when 9 2 2 , o r t h e boolean s e t PS, when q
r-k =(j-k~
(lb) B(k,j,r)
( l b ' ) B(k,k+l,r) ( l c ) ;(k,r)
=
Moreover, i f c o n d i t i o n ( 1 (2) f
=
f
q'
for all
= f r-k)
F(r-k)
a(k,r)
D < k.S r
for all
( l a ) a ( k , r ) = {;If
1);
c r k M;
0 6 k < ;i < r s r k M;
for all
for a l l
=
O s k i r s r k M;
O < k < r < r k M.
i s s a t i s f i e d , then r k q'
= [ ]
l3(k,j,r)
=
[;][I,,
;(k,r)
= [r-kl!
P r o o f . I t i s w e l l known ( c f . [ 3 ] , [ 1 6 ] ) t h a t ( 1 ) i m p l i e s ( l a ) - ( l c ) and ( 2 ) . (la)-(l).
By Prop. 2.2,
we g e t
Ar,l r r a ( 2 , r ) = A* {21f -- 121f
231
s
P.V. Ceccherini and A. Sappa
148
(Ib')
=,
( 1 ) . By Prop. 2.2,
we g e t R ( k , k t l , r )
=
'r k 'ktl
4
r,k
=
'k+l,k'
.
f o r k = l we o b t a i n a(k,r)
A
= A
r,l
= R(O,k,rl
f(r-k)
= f(r-k)
and ( 1 ) f o l l o w s as above.
2,l
r-0 I k-O'f
=
,k
r - 'k'f' -
(lb)
=,
(la).
(Ic)
=)
( l b ) . I n each maximal c h a i n of f l a t s between a k - f l a t A and a r - f l a t
B t h e r e e x i s t s e x a c t l y one j - f l a t C ( 0 k ~c j < r ( r k M ) ; g i v e n such a C, t h e j o i n -
i n g of a maximal c h a i n between A and C and o f a maximal c h a i n between C and El i s a maximal c h a i n between A and B. {(C,$):C
between A and B), g i v e s :
i s a j - f l a t belonging t o a chain 0
F(r-k)
Nk,j,r)
=
REMARK 2.4. define
The double c o u n t i n g argument, a p p l i e d t o t h e s e t
a(k,r),
F(j-k)F(r-j),
B(k,j,r)
- {r-k j-kff.
F(r-k)
= F(r-jlF(j-kl
0
L e t M be t h e m a t r o i d o f f l a t s o f an a f f i n e space o f o r d e r q and R(k,j,r)
+
and v ( k , r ) as i n Prop. 2.2. From ( 2 ) o f Prop. 2.3,
f o l l o w s t h a t f ( 0 ) = 0, f ( t ) = qt-', r-k a(k,r) = q REMARK 2.5. such t h a t
i.e.
(k,r)
F ( t ) = qt(t-1)'2
r-1
[k-llq, B ( k , j , r )
=
[;:;Iq,
;(k,r)
it
and f o r k > O = [r-kl
q
!.
We g i v e now an o t h e r example o f a PMD M with s i z e - f u n c t i o n f
# F ( r - k ) . L e t S be a S t e i n e r system S(Z,k,n)
(e.g. t h e S t e i n e r
system whose b l o c k s a r e t h e l i n e s o f a p r o j e c t i v e ( r e s p . a f f i n e ) space S o f o r d e r q w i t h k = q t l ( r e s p . k = q ) ) and l e t M be t h e PMD ( o f r a n k 3 ) whose f l a t s a r e t h e empty s e t ,
the s i n g l e points,
t h e b l o c k s and t h e f u l l s e t o f p o i n t s . The s i z e
f u n c t i o n f i s such t h a t f ( 0 ) = 0, f ( 1 ) = 1, f ( 2 ) = k, f ( 3 ) = n; t h e a s s o c i a t e d f u n c t i o n F i s such t h a t F(0) = F ( 1 ) = 1, F ( 2 ) = k, F ( 3 ) = nk; moreover =
= 1,
A
2,l
= k-1,
A
3,l
=-
'-' . k
=
Therefore
2 whenever n f k - k + l ( i n t h e p r e v i o u s examples, whenever t h e space S i s n o t a p r o j e c t i v e plane).
149
F-Binomial Coefficients and Related Combinatorial Topics 3. POSETS OF FULL
BINOMIAL TYPE
These s t r u c t u r e s have been i n t r o d u c e d i n [111, ( c f . a l s o [71). We s h a l l use t h e n o t a t i o n o f $1.
(P,*.) be a p a r t i a l l y o r d e r e d s e t ( p o s e t ) w i t h minimum 0, and l e t a,b be elements o f P. I f a 4 b, t h e i n t e r v a l between them i s d e f i n e d by I ( a , b ) = Let
=
f c ~ P :a < c c b l . I f a < b a maximal c h a i n between a and b i s a c h a i n
a = a < a <...< an = b o f I ( a , b ) which i s n o t i n c l u d e d i n a l o n g e r c h a i n o f 0 1 I ( a , b ) ; t h e number n i s c a l l e d t h e l e n g t h o f t h e c h a i n . We s h a l l denote by ;(a,b) t h e number o f maximal c h a i n s o f I ( a , b ) , We s h a l l say t h a t (JD)
and we s h a l l assume
;(a,b)
=
1 when a=b.
P i s a JD-poset i f i t s a t i s f i e s t h e f o l l o w i n g c o n d i t i o n
(Jordan-Dedeking c h a i n c o n d i t i o n ) : i f a,b
E
f
w i t h a < b, a l l t h e maximal
c h a i n s o f I ( a , b ) have one and t h e same l e n g t h , denoted by d(a,b)
and c a l l e d t h e
d i s t a n c e between a and b ( o r t h e l e n g t h o f t h e i n t e r v a l ( I ( a , b ) ) . If a=b, we p u t d(a,b)
=
If
0.
P i s a JD-poset w i t h minimum 0, t h e n t h e rank o f an element a
d e f i n e d by r k a=d(O,a), and t h e rank o f We n o t e t h a t i f a,bEP
there exists a function F : f O , l , ;(a,b)
interval o f length n o f that
i s d e f i n e d by r k
w i t h a < b, t h e n d(a,b)<
We s h a l l say t h a t a JD-poset a < b, t h e number
P
rk
P
P
P.
P ( w i t h minimum 0 ) has a c h a i n f u n c t i o n if
...,r k
PI
+
N such t h a t , f o r a l l a,b E 2 w i t h i.e.
every
has F ( n ) maximal c h a i n s . I n t h i s case we s h a l l say
p i s an F-chain p o s e t . A poset
DEFINITION 3.1. (1)
P
i s c a l l e d a poset o f f u l l b i n o m i a l t y p e i f
P has minimum 0,
(2) P
i s a JD-poset,
(3)
P i s a F - c h a i n poset, f o r some F.
If
f i s a JD-poset w i t h minimum 0, f o r each i n t e r v a l I ( a , b )
each i n t e g e r k, w i t h O s k s d ( a , b ) , Ik(a,b)
= I c E I(a,b)
We s h a l l say t h a t f : IO,1,
...,r k
+ .
P and f o r
of
we s h a l l p u t
: d(a,c) = k)
P has a s i z e f u n c t i o n _ i f t h e r e e x i s t s a f u c n t i o n
P)
is
= max t r k a: a € $ , .
o f maximal c h a i n s between them i s F ( d ( a , b ) ) ,
P
E
N such t h a t , f o r a l l i n t e r v a l s I ( a , b ) o f f ,
f,
I50
P.V. Ceccherini and A . Sappa IIl(a,b)I
= f(d(a,b)).
I n t h i s case we s h a l l say t h a t
P
i s an f - s i z e poset. The f o l l o w i n g p r o p o s i -
t i o n y i e l d s o t h e r e q u i v a l e n t d e f i n i t i o n s o f poset o f f u l l b i n o m i a l t y p e . PROPOSITION 3.2.
P
Let
be a JD-poset w i t h minimum 0; t h e f o l l o w i n g c o n d i -
t i o n s a r e e q u i v a l e n t , where f and F a r e m u t u a l l y a s s o c i a t e d f u n c t i o n s : (a)
( b l IIk(a,b)I (c)
P
P i s an F-chain poset ( i . e .
i s a poset o f f u l l binomial type).
(Oc k c d(a,b)).
= t d(a,b)]
P i s an f - s i z e poset.
Proof. Let a , b E P , (a) * ( b ) . e x a c t l y one c E
acb.
L e t be O < k < d ( a , b ) .
I n each maximal c h a i n o f I ( a , b ) t h e r e e x i s t s
P such t h a t d(a,c)
= k.
I f c i s such an element o f I ( a , b ) ,
the
j o i n i n g o f a maximal c h a i n o f I ( a , c ) and o f a maximal c h a i n o f I ( c , b ) i s a maximal c h a i n o f I ( a , b ) . CL
The double c o u n t i n g argument a p p l i e d t o t h e s e t I ( c , o ) :
i s a maximal c h a i n o f I ( a , b ) , F(d(a,b)) = IIk(a,b)
I
d(a,c)
= k)
C E ~ ,
gives:
F ( k ) F(d(a,b)-k),
so t h a t ( b ) f o l l o w s .
( b ) * ( c ) . For k = l we o b t a i n I1 ( a , b ) I = I d(;yb))f = f ( d ( a , b ) . 1 ( c ) * ( a ) . A c t u a l l y I 1 ( a , b ) I = f ( d ( a , b ) ) . We now a p p l y i n d u c t i o n on d ( a , b ) ; 1 i f d(a,b) = 0,1, t h e n 1 = ;(a,b) = F ( d ( a , b ) ) . I f d(a,b),2, a p p l y t h e double coun t i n g argument t o t h e s e t { ( c , ~ ) : c E
U,U
i s a maximal c h a i n o f I ( a , b ) ,
d(a,c)=ll.
By t h e p r e v i o u s argument and b y t h e i n d u c t i o n h y p o t h e s i s , we have ;(a,bl
= (Il(a,b)I
EXAMPLE 3.3.
F(1) F(d(a,bl-l)
= f(d(a,b))
P= 2
I f X i s a f i n i t e s e t and i f
o f X o r d e r e d by i n c l u s i o n , t h e n
P
F(d(a,b)-l)
X
= F(d(a,bl).
i s the set o f a l l
0
subsets
i s a poset o f f u l l b i n o m i a l type, w i t h s i z e
f u n c t i o n f ( t ) = t and c h a i n f u n c t i o n F ( t ) = t ! ( 0 t ~c
1x1)
any such f u n c t i o n s f and F ( w i t h dom f = dom F = t O , l ,
...,n l ) ,
(cf.
[ 1 3 1 ) . Conversely
( c f . example 1.21, X can be c o n s i d e r e d as s i z e f u n c t i o n and c h a i n f u n c t i o n o f a boolean s e t p = 2 w i t h 1x1 = n. We n o t e t h a t , i f P = I X I U ( ox ) U ( ,x) U...U ( Xk ) w i t h 2 4 k < 1x1 - 1, t h e n t h e
P i s n o t of f u l l b i n o m i a l t y p e : i f Y
poset
= d(Z,X)
z
to
x.
= 2, b u t t h e r e a r e
X
E ( 2 ) and
2 c h a i n s from 0 t o Y and
Z E(krl)l
1x1
-
t h e n d(0,Y)
=
(k-1) > 2 chains from
F-Binomial Coefficients and Related Combinatorial Topics EXAMPLE 3.4. dimension n-1
IS1
If X i s a f i n i t e p r o j e c t i v e space PG(n-1.q)
o f o r d e r q and
2 and i f ?' i s t h e s e t o f f l a t s o f X o r d e r e d by i n c l u s i o n , t h e n ?
i s a poset o f f u l l b i n o m i a l t y p e w i t h s i z e f u n c t i o n f ( t ) = [ t ]
and c h a i n f u n c t i o n q Conversely, any such f u n c t i o n s f = [ ] and
F ( t ) = [ t ] !. ( c f . [ 3 ] , prop. 3.3.). 9 h F = [ I ! w i t h q = p ( h h l , p p r i m e 3 2 1 and dom f = d a n F = 10.1
,...,n l
f-l
q
( c f . exaE
p l e 1 . 3 ) can be c o n s i d e r e d as t h e s i z e and t h e c h a i n f u n c t i o n s o f t h e poset o f t h e subspaces of a p r o j e c t i v e space X = PG(n-l,q) We n o t e t h a t , if with i
=
P
o f o r d e r q and dimension n-1.
i s t h e s e t of a l l t h e i - d i m e n s i o n a l f l a t s o f X = PG(n-l,q),
-l,O,...,k,n-1
where 1 4 k < n - 2 , t h e n t h e poset
P i s not o f f u l l binomial
t y p e : i f Y i s a l i n e and 2 i s a ( k - 1 ) - d i m e n s i o n a l f l a t , t h e n d(0,Y) b u t t h e r e a r e q + l c h a i n s f r o m 0 t o Y and q wk-' +
EXAMPLE 3.5. I f
$=
...+q+l > q + l
,,..., xla ,..., xnl ,...,x na 1
{xo,xl
elements (a,n a l ) , o r d e r e d by assuming xo as minimum and x
= d(Z,X)
= 2
c h a i n s f r o m Z t o X.
i s any s e t w i t h a n t 1 ij
< x
hk
i f f i < h, t h e n
2 i s a p o s e t o f f u l l b i n o m i a l t y p e w i t h s i z e f u n c t i o n f ( O ) = O , f ( l ) = l , f ( t ) ad with t-1 chain functionF(O)=F(l)=l,F(t) = a ( 2 6 t < n ) . Conversely, any such f u n c t i o n s f and F ( c f . example 1.3.) can be c o n s i d e r e d as t h e s i z e and t h e c h a i n f u n c t i o n s o f a poset o f f u l l b i n o m i a l t y p e p as above. REMARK 3.6.
L e t M be t h e PMD o f t h e f l a t s o f an a f f i n e space X = AG(n-1,q) and l e t B
o f dimension n - 1 3 2 ,
not o f f u l l binomial type: t h e n d(0,Y)
z
to
= d(Z,X)
be t h e p o s e t o f t h e f l a t s o f X. The poset
P
is
i f Y i s a l i n e and Z i s an ( n - 3 ) - d i m e n s i o n a l f l a t ,
= 2 , b u t t h e r e a r e q c h a i n s f r o m 0 t o Y and q + l c h a i n s f r o m
x. Therefore,
if
M i s a PMD on a f i n i t e s e t X and if P i s t h e s e t of t h e f l a t s
o f M ordered by inclusion, then
P i s not necessarily a poset o f f u l l binomial
t y p e : t h e number o f maximal c h a i n s i n an i n t e r v a l I ( a , b ) does n o t depend o n l y on t h e l e n g t h o f t h e i n t e r v a l , b u t a l s o on t h e r a n k s o f a and b ( c f . a l s o Remark 2.5,
and Examples 3.3,
3.4).
The case when
P i s a poset o f f u l l b i n o m i a l t y p e
i s characterized by t h e f o l l o w i n g proposition. PROPOSITION 3.7.
L e t M be a PMD on a f i n i t e s e t X and l e t
t h e f l a t s o f M o r d e r e d b y i n c l u s i o n . The poset
P
P
be t h e poset o f
i s of f u l l b i n o m i a l t y p e i f and
o n l y i f M i s one o f t h e f o l l o w i n g PMD's: ( a ) M i s t h e t r i v i a l PMD o f r a n k 1 o r t h e t r i v i a l PMD o f r a n k 2 on X ( i . e .
M i s a t r i v i a l g r a p h i c space o f dimension 0 on 1 on X r e s p . ) ;
152
P.V. Ceccherini and A . Sappa (b) M i s t h e m a t r o i d 2
X
o f a l l subsets o f X, i . e . M i s t h e g r a p h i c space o f
o r d e r 1 and dimension 1x1-1;
( X I o f a l l f l a t s o f a p r o j e c t i v e space, o f dimenY q h a v i n g X as s e t o f p o i n t s .
( c ) M i s t h e m a t r o i d Mn-l s i o n n-1 and o r d e r q * 2 ,
P r o o f . I f M i s a PMD as i n ( a ) - ( c ) , t h e n t h e poset
P
o f i t s f l a t s i s a poset
o f f u l l b i n o m i a l t y p e ( c f . Ex. 3.3 and Ex. 3.4). L e t M b e a PMD on X such t h a t t h e p o s e t b i n o m i a l type. I f r k
M r 2,
B o f i t s f l a t s i s a poset o f f u l l
t h e n we a r e i n t h e case ( a ) . If r k
M > 2,
l e t f denote
t h e s i z e f u n c t i o n o f M. F o r any f l a t Y o f r a n k 2, we have t h a t lIl(O,Y)l
P
I n o t h e r words, t h e p o s e t
= ( Y I = f ( 2 ) = f(d(0,Y)).
has t h e same s i z e f u n c t i o n f t h a n t h e PMD M. Thus
c o n d i t i o n ( b ) o f Prop. 3.2 holds; i t means t h a t c o n d i t i o n ( l b ) o f Prop. 2.3 h o l d s . So c o n d i t i o n ( 1 ) o f Prop. 2.3 h o l d s too, and t h e r e s u l t ( b ) - ( c ) i s proved.
0
4. F-GEODETIC GRAPHS a l l graphs w i l l be f i n i t e w i t h o u t l o o p s o r m u l t i p l e edges,
I n what f o l l o w s ,
an:i a l l d i r e c t e d graphs w i l l be w i t h o u t d i r e c t e d c i r c u i t s . Any d i r e c t e d graph
p =
p(t) =
G'
(V,;)
=
i s o b v i o u s l y t h e Hasse diagram o f a poset
where x < y i f and o n l y if t h e r e e x i s t s a d i r e c t e d p a t h f r o m x
(V,<)
t o y; c o n v e r s e l y t h e Hasse diagram o f a poset p = (V,<)
6
= (V,:),
=
that
E
where ( x , y )
P(E) a r e
and
I f G = (V,E)
EE
i s a d i r e c t e d graph
i f and o n l y i f x i s covered by y. We s h a l l say
mutually associated.
(resp.
6
*
= (V,E))
i s a graph ( r e s p . d i r e c t e d graph) and i f two
v e r t i c e s x,yE V a r e j o i n e d by a p a t h ( r e s p . d i r e c t e d p a t h ) , t h e d i s t a n c e d ( x , y ) ( r e s p . i ( x , y ) ) i s d e f i n e d as t h e number o f edges i n a geodesic i . e . p a t h (resp. d i r e c t e d s h o r t e s t p a t h ) between x and y. We denote by (resp. b y
*
r (x,y)
+
= rG(x,y))
i n a shortest r(x,y) =rG(x,y)
t h e s e t o f d i s t i n c t geodesics o f G (resp.
of
between x and y; we p u t : v(x,y) q(x,x)
=
I
r(x,y)l
y(x,x)
= 1
diam G = max I d ( x , y )
and ;(x,y) and
=
;I
x,y)I +
d ( x , x ) = d x,x)
: x,y E
Vj,
d am
We s h a l l say t h a t a connected graph G =
E
V,E)
ifx # y; = 0;
= max ~+d(x,y) : x,y E W .
( r e s p . a d i r e c t e d graph
5
= (V,:))
El
153
F-Binomial Coefficients and Related Combinatorial Topics
...,diam
i s a graph w i t h a geodetic f u n c t i o n i f t h e r e e x i s t s a map F : ( O , l , (resp. F : ( O , l ,
*
...,diam
G)
N ) such t h a t u ( x , y ) = F ( d ( x , y ) ) ( r e s p .
+
+
GI
+
N
u(x,y) =
i) i s
= F(J(x,y))
f o r a l l x,y E V .
F-geodetic.
Note t h a t F(0) = 1 and t h a t F(n) # 0 f o r a l l n E dom F, so t h a t t h e as -
I n t h i s case we s h a l l a l s o say t h a t G (resp.
sociated f u n c t i o n f can be considered as i n $1. Let
2
= (V,;)
be a d i r e c t e d graph and l e t x , y ~ V be such t h a t x - < y . Then t h e
i n t e r v a l I ( x , y ) i s defined by I(x,y) := I z E V :
+
x < z - ( y l , i . e . I ( x , y ) i s defined as i n P ( G ) ,
and t h e geodetic i n t e r v a l I g ( x , y ) i s d e f i n e d by P(x,y)
:= t z e V:
Note t h a t I 9 (x,y)
C;(x,y) f o r some
ZE
5 I(x,y)
C; E r ( x , y ) ~ .
and t h a t I g (x,y) = t z e I ( x , y ) : i ( x , z )
= ;(x,y)-;(y,z)).
For any 1 < k < i ( x , y ) , l e t : = I z e I g ( x , y ) : ~ ( x , z )= k l
I;(x,y)
E
We say t h a t
= (V,:)
has a source
= (zE
I9 ( x , y ) : J ( Z , Y ) = i ( x , y ) - k l .
O E V i f f o r any
XE
V \ (01 t h e r e e x i s t s a d i r e c -
t e d path from 0 t o x.
A graph
6
= (V,i)
w i l l be c a l l e d a d i r e c t e d graph o f f u l l binomial t y p e ( w i t h
geodetic f u n c t i o n F ) i f : (a)
E
has a source 0,
( b ) f o r a l l x , y ~ Vw i t h x < y : (c)
Let
E
be t h e poset associated w i t h
(2)
= I(x,y),
G i s F-geodetic f o r some F.
PROPOSITION 4.1.
(1)
I 9 (x,y)
P has a minimum P i s a JO-poset
= (V,i)
6
be a d i r e c t e d graph and l e t P = P ( 6 ) = ( V , < )
(so t h a t
=
0 i f and o n l y i f
6(P)).
Then
has source 0;
i f and o n l y i f I g ( x , y )
I(x,y),
=
for a l l x , y ~ V
with x
6
P
i s a poset o f f u l l binomial type w i t h chain f u n c t i o n F i f and o n l y i f
i s a d i r e c t e d graph o f f u l l binomial type w i t h geodetic f u n c t i o n F. Proof. -
( 1 ) i s obvious. ( 2 ) :
s e t M(x,y) o f maximal chains o f G(x,y;of
E
P i s a JD-poset P i n I ( x , y ) i s the
O f o r a l l x , y ~ Vw i t h x c y
0
f o r a l l x , y ~ Vw i t h x < y t h e set
I(x,y) = Ig(x,y).
? ( x , y ) o f t h e geodesics
( 3 ) : P i s a poset
of
P.V . Ceccherini and A . Sappa
154
f u l l binomial type w i t h chain f u n c t i o n F
I
(M(x,y) Ig(x,y)
F(+d(x,y)) f o r a l l x < y i n V
=
= I(x,y)
and [;(x,y)(
= F(d(x,y))
t y p e w i t h g e o d e t i c f u n c t i o n F. The poset
p = (V,<),
o
O P has minimum 0, I' i s a JD-poset and
"G
has source 0, f o r a l l x < y i n V
E
i s a d i r e c t e d graph o f f u l l b i n o m i a l
0
where V = IO,x,y,z,tl
and O < x < z < y , O < t < y , i s n o t a
b u t t h e graph 6 ( p ) i s F-geodetic ( w i t h F = l ) ; n o t e t h a t Ig(O,y)
JD-poset, = {O,t,Yl
c I(0,y)
=
=
v.
PROPOSITION 4.2.
Let
"G
=
be a d i r e c t e d graph. The f o l l o w i n g c o n d i t i o n s
(V,i)
a r e e q u i v a l e n t (where F and f a r e m u t u a l l y a s s o c i a t e d ) : i s F-geodetic f o r some F,
(a)
( b ) f o r a l l x < y i n V and f o r a l l 1 g k c a ( x , y ) : (c) f o r a l l x < y i n V:
II~I=
*
(c)
=)
=
+
1
t o t h e s e t Z1 g i v e s : = F(a(~,y)).
If
2
ated t o
E.
gives:
F.
1.
( a ) . We a p p l y i n d u c t i o n on a ( x , y ) 3 1 . I f a ( x , y )
= F(i(z,y))
z(x,z) = k l
+
9 d ( x ,y so t h a t I k ( x , y ) = I
= 1 = F ( 1 ) . Suppose d ( x , y ) h 2 ; by t h e i n d u c t i o n hypothesis,
IT(z,y)
J(X,Y) k IF'
The double c o u n t i n g argument
(z,<(x$y)): z ~ < ( x , y ) ,ger(x,y),
=
F ( d ( x , y l ) = I I z ( x , y ) l F ( k ) F(a(x,y)-k), (b) * (c) for k
k
f(i(x,y)).
Proof, L e t x,y be elements o f V w i t h x < y . ( a ) - ( b ) . a p p l i e d t o t h e s e t Zk
9
( I (x,y)I = i
1
I
=
we have 1 Therefore t h e double c o u n t i n g argument a p p l i e d
= F(G(x,y)-l).
I?(x,y)
1, t h e n I;(x,y) when Z E Z
= IIg(x,y) l F ( l ) F ( i i ( ~ , y ) - l ) = f ( a ( x , y ) )
1
F(a(x,y)-l)=
0
i s a d i r e c t e d graph, we s h a l l denote by G t h e u n d i r e c t e d graph a s s o c i -
PROPOSITION 4.3.
Let
= (V,:)
be a d i r e c t e d graph w i t h source O S V . Suppose
t h a t f o r each x E V t h e p a r i t y o f t h e l e n g t h o f any d i r e c t e d path f r o m 0 t o x depends o n l y on x; w r i t e p ( x ) =O i f i t i s even and p ( x ) = 1 i f i t i s odd. Then +
t h e u n d i r e c t e d graph G a s s o c i a t e d t o G i s connected and b i p a r t i t e . P r o o f . G i s connected s i n c e 0 i s a source. We v e r i f y t h a t G i s b i p a r t i t e by 0 1 i assuming V = V U V where V = I x E V : p ( x ) = il, i = 0 , l . We have t o prove t h a t i f ( x , ~ ) EE then p ( x ) # p ( y ) . We can suppose w i t h o u t loss o f g e n e r a l i t y t h a t (X,Y)E
E.
I f G(0,x) and G(0,y)
a r e geodesics f r o m 0 t o x and t o y resp.,
then
F-Binomial Coefficients and Related Combinatorial Topics
155
g(0,y) and G(0,x) u (x,y) are both d i r e c t e d paths from 0 t o y, so t h a t t h e i r lengths have t h e same p a r i t y p ( y ) . It f o l l o w s t h a t p ( x ) # p ( y ) . PROPOSITION 4.4.
L e t G = (V,E) be a connected b i p a r t i t e graph and l e t 0 be
any vertex o f G. Then by s t a r t i n g from 0, a n a t u r a l o r i e n t a t i o n can be d e f i n e d
on E, i n such a way t h a t
= (V,:)
i s a d i r e c t e d graph w i t h source 0 (and w i t h o u t
d i r e c t e d c i r c u i t s ) . It f o l l o w s t h a t
$=
where x
E
Proof. I f x,y E V w i t h d(0,x)
-*
p(G)
=
(V,C) i s a poset w i t h minimum 0,
a d i r e c t e d path from x t o y .
= d(O,y),
then x and y cannot be adjacent be-
cause G i s b i p a r t i t e . Let ( x , y ) be an edge o f G. We have e i t h e r d(0,y) = d(O,x)+l o r d(O,x)
= d(0,y)tl.
Orient t h e edge from x t o y i n the f i r s t case, from y t o +
x i n t h e second. It i s easy t o show t h a t , whenever t h e r e i s an edge (x,y) EE, then (x,y)
i s t h e o n l y d i r e c t e d path from x t o y, and t h a t whenever t h e r e e x i s t s
a d i r e c t e d path from x t o y then t h e r e i s no d i r e c t e d path from y t o x; indeed i s a d i r e c t e d path, then we have
(by i n d u c t i o n on i ) , i f (xo,x l,...,xi) (xo,xl)
,..., ( X ~ - ~1 , X . and ) E ~d(O,xi)
THEOREM 4.5.
= d(O,xo) t i.
0 +
+
L e t G = (V,E) be a connected b i p a r t i t e graph and l e t G = (V,E)
be the d i r e c t e d graph obtained by s t a r t i n g from a given vertex OEV as i n Prop.
4.4. Then, whenever x and y are elements o f
V w i t h x < y , the f o l l o w i n g c o n d i t i o n s
are e q u i v a l e n t : (a
p(x,y) i s a d i r e c t e d path from x t o y i n
(b
p(x,y)
(C
p(x,y) i s a geodesic from x t o y i n G.
Proof.
t,
i s a geodesic from x t o y i n 6, *
be any d i r e c a) * ( b ) . Assume x = 0 f i r s t . L e t p(0,y) = (O=x",,.,,x.=y) 1
t e d path o f lenght i from x t o y i n
5.
We have d(O,y)=i, by t h e i n d u c t i o n argu-
ment sketched a t the end o f t h e p r o o f o f Prop. 4.4.
Therefore p(0,y) = p ( x , y ) i s
a geodesic i n G. Assume now 0 < x < y . L e t p(0,x)
and p ( x , y ) be any d i r e c t e d paths i n
8
from
0 t o x and from x t o y resp. By g l u e i n g p(0,x) and p(x,y) we get a d i r e c t e d path p(0,y)
in
'G.
For t h e previous case p(0,y)
i s a geodesic i n G. Thus i t s subpath
p(x,y) must a l s o be a geodesic i n 6.
(b)
=)
( a ) . I n d u c t i o n on i = d ( x , y ) .
x < y . so t h a t p(x,y)
When i = l , we have p(x,y) = ( x , y ) ~ isince *
i s a d i r e c t e d path i n G. Assume now i Z 2 and suppose t h a t
P.V. Ceccherini and A . Sappa
156
any geodesic o f G o f l e n g t h j w i t h 1 6 j < i i s a d i r e c t e d p a t h i n
,...,x 1. - 1
p ( x , y ) = ( x =x,x
,xi=y)
and ( X ~ - ~ , Xo f~ G. ) These a r e b o t h d i r e c t e d p a t h s i n Then p ( x , y ) i s a d i r e c t e d p a t h i n (c) *(a)
obviously.
(b)*(c).
I f p(x,y)
6.
The geodesic
i s o b t a i n e d by g l u e i n g t h e geodesics p(x.xi-,)
6
by t h e i n d u c t i o n h y p o t h e s i s .
E.
...,x .1= y ) i s a geodesic i n G, t h e n p(x,y) i s 0 1' because (because ( b ) = . ( a ) ) , and i t must be a geodesic i n
a d i r e c t e d path i n
= ( x =x,x
6,
+
any s h o r t e r d i r e c t e d p a t h p ' ( x , y )
i n G would be a p a t h o f G s h o r t e r t h a n p ( x , y ) ,
which i s i m p o s s i b l e s i n c e p ( x , y ) i s a geodesic i n G. COROLLARY 4.6. L e t G
(V,E)
=
0
be t h e
be a connected b i p a r t i t e graph and l e t
d i r e c t e d graph o b t a i n e d by s t a r t i n g f r o m a g i v e n v e r t e x O E V as i n Prop. 4.4. Then ( 1 ) whenever x , y ~ V w i t h x r y , we have d ( x , y ) that
= +d(x,y), r ( x , y )
= ?(x,y)
so
r(x,y) = t(x,y); ( 2 ) G i s F-geodetic i f and o n l y i f
Proof.
E
i s F-geodetic f o r a l l O E V .
( 1 ) i s obvious. For ( 2 ) i t i s enough t o n o t e t h a t ;(x,y)
= r ( x , y ) when we
assume 0 = x. COROLLARY 4.7.
Let G
=
be a connected b i p a r t i t e graph. Then t h e f o l -
(V,E)
l o w i n g c o n d i t i o n s a r e e q u i v a l e n t , where F and f a r e m u t u a l l y a s s o c i a t e d f u n c t i o n s : ( a ) G i s F-geodetic,
IIzEV:
d(x,y)
( b ) x,yEV,
Otkcd(x,y)*
( c ) x,yEV
*
REMARK 4.8.
The statement o f C o r o l l a r y 4.7 also h o l d s i f G i s n o t b i p a r t i t e ;
~ [ Z E V :d(x,z)
d ( x , z ) = k, d(z,y)
= 1,
d(z,y)
= d(~,yl-kl(=
d(x,y)-llI
I
= f(d(x,y)).
)f,
0
i t can be proved by a d i r e c t argument ( c f . [ 1 6 1 ) . T h i s can a l s o be o b t a i n e d f r o m
t h e p r o o f o f Prop. 4.2, r(x,y), (zEV:
d(x,y)
-t
by r e p l a c i n g g(x,y),
+
r(x,y),
+
d(x,y)
(and by exchanging consequently t h e s e t I!(x,y)
r e s p . w i t h g(x,y), and t h e s e t
d ( x , z ) = k, d ( z , y ) = d ( x , y ) - k ) ) .
EXAMPLE 4.9.
The complete graph Kn, a t r e e G , t h e ( 2 k t l ) - c i r c u i t G a r e F-geo-
d e t i c graphs w i t h F = l . An F-geodetic graph w i t h F-1 i s c a l l e d g e o d e t i c - g r a p h ( c f . [161, [ l a ] ) . The i l k - c i r c u i t G i s F - g e o d e t i c w i t h F ( t ) = l , f o r t < k , and w i t h F ( 2 ) = 2 otherwise. EXAMPLE 4.10.
The complete b i p a r t i t e graph Kn,,=(V,E)
with V =
V'UV",
157
F-Binomial Coefficients and Related Combinatorial Topics
IV'(
=
IV"I
= n i s F-geodetic w i t h F ( 0 ) = F ( 1 ) = 1, F ( 2 ) = n.
A hypercube i s an F-graph w i t h F ( t ) = t ! . Conversely any con-
EXAMPLE 4.11.
131).
n e c t e d b i p a r t i t e F-graph w i t h F ( t ) = t ! i s a hypercube ( c f .
(cf. (51,
= t (0
The graph Kn x Km i s F - g e o d e t i c w i t h F ( t
EXAMPLE 4.12.
< t s 2 cn,m)
[151 1 -
I f Qn i s t h e n-cube, t h e graph Qn x K
EXAMPLE 4.13.
s
m
F-geodetic
with
F ( t ) = t ! ( O s t < n t l ) ( c f . [51, Prop. 3.1).
i s F - g e o d e t i c f o r some F
I f G i s a connected graph and i f GxK,
EXAMPLE 4.14.
and some m x 2 , t h e n F ( t ) = t ! . Moreover, i f G i s b i p a r t i t e , t h e n G i s a hypercube (and GxK i s a l s o a hypercube i f and o n l y i f m = 2 ) ( c f . [ 5 ] , Prop. 3.1, Cor. 3.2) m EXAMPLE 4.15. L e t G be t h e d i r e c t e d graph whose v e r t i c e s a r e t h e subspaces ~o f a g r a p h i c space o f dimension n and o f o r d e r q 3 1 and ( x , y f l a t x i s covered b y t h e f l a t y. Then
6
i s F-geodesic w i t h F ( t
i s an edge i f t h e =
[tlq!( c f .
[31,
Prop. 3.3).
REMARK 4.16.
E
L e t G be t h e u n d i r e c t e d graph a s s o c i a t e d t o t h e d i r e c t e d graph
c o n s i d e r e d i n t h e Ex. 4.15.
I n o t h e r words, G i s t h e q-analogue o f Qn-,.
Note
t h a t when q a 2, G i s n o t F-geodesic: i f x,y a r e two p o i n t s and z i s a p l a n e cont a i n i n g x, t h e n d(x,y)
= d(x,z)
t h a t t h e d i r e c t e d graph
5
graph o f Example 4.15;
=
2, b u t 2 = v(x,y)
#
u ( x , z ) = q t l . Note a l s o
o b t a i n e d f r o m G by s t a r t i n g f r o m t h e empty f l a t i s t h e
when we s t a r t from a f l a t O f 0 , t h e n
E
i s F - g e o d e t i c i f and
only i f q=l. EXAMPLE 4.17. o f Ex, 3.5.
Then
Let
6
6
=
6(P) be
t h e d i r e c t e d graph a s s o c i a t e d t o t h e p o s e t
p
i s F-geodetic a c c o r d i n g l y t o Prop, 4.1 ( b u t G i s n o t F-geode-
tic). ACKNOWLEDGEMENT.
T h i s r e s e a r c h was p a r t i a l l y supported by GNSAGA o f CNR and by
MPI.
BIBLIOGRAPHY [ 1 1 F. Buekenhout, Une c h a r a c t e r i z a t i o n des espaces a f f i n s basee s u r l a n o t i o n de d r o i t e , Math. Z. 111 (1969) 367-371. [ 21 P . V . C e c c h e r i n i , 78-98.
S u l l a nozione d i s p a z i o g r a f i c o , Rend.
Mat.
( 5 ) 6 (1967)
P.V , Ceccherini and A . Sappa
158
[ 3 ] P.V. C e c c h e r i n i , A q-analogous o f t h e c h a r a c t e r i z a t i o n o f hypercubes as graphs, J. Geometry 22 (1984) 57-74. [ 4 1 P.V. Ceccherini, A. Dragomir, Combinazioni g e n e r a l i z z a t e , q - c o e f f i c i e n t i b i n o m i a l i e spazi g r a f i c i , A t t i Convegno Geometria Combinatoria e sue a p p l i c a z i o n i (Peruaia. Settembre 1970) 137-158.
-~
[ 5 1 P.V.
Ceccherini, A. Sappa, A new c h a r a c t e r i z a t i o n o f hypercubes,Annals D i s c r e t e Math. ( t h i s volume).
[6
1 L.
[7
1 L.
C e r l i e n c o , F. P i r a s , C o e f f i c i e n t i b i n o m i a l i g e n e r a l i z z a t i , Fac. S c i . C a g l i a r i 52 (1982) 47-56.
Rend.
Sem.
C e r l i e n c o , F. Piras, G-R-Sequences and i n c i d e n c e coalgebras o f p o s e t s o f f u l l b i n o m i a l t y p e , ( t o appear).
[ 8 ] R.J. Cook, D.G. [ 9 ] H. Crapo, G.C.
Pryce, U n i f o r m l y g e o d e t i c graphs, t o appear. Rota, C o m b i n a t o r i a l geometries, MIT Press, Cambridge (1970).
[ 1 0 1 M. Deza, N.M. S i n g h i , Some p r o p e r t i e s o f p e r f e c t m a t r o i d designs, Annals D i s c r e t e Math. 6 (1980) 57-76. [ 11 ] P. D o u b i l e t , G.C. Rota, R.P. SRanley, On t h e f o u n d a t i o n s o f C o m b i n a t o r i a l t h e o r y V I : t h e i d e a o f g e n e r a t i n g f u n c t i o n , i n G.C. Rota ( e d . ) , F i n i t e Oper a t o r Calculus, Academic Press, New York (1975) 83-134. [ 12 1 J. Edmonds, U.S.R. Murti, P. Young, E q u i c a r d i n a l m a t r o i d s and m a t r o i d d e s i gns, i n "Combinatorial Mathematics and i t s A p p l i c a t i o n s " , Second Chapel H i 11 Conference ( 1 9 7 0 ) .
[ 1 3 1 S. Foldes, A 159.
c h a r a c t e r i z a t i o n o f hypercubes, D i s c r e t e Math. 17 (1977) 155-
[ 14 1 B.L. R o t h s c h i l d , N.M. S i n g h i , C h a r a c t e r i z i n g k - f l a t s i n geometric designs, J. Comb. Theory A 20 (19761, 398-403. [ 15 1 A. Sappa, C a r a t t e r i z z a z i o n e d i g r a f i t r a m i t e geodetiche, Tesi, Univ. d i Roma, D i p a r t i m e n t o d i Matematica (1984). [ 16
I R.
S c a p e l l a t o , On g e o d e t i c graphs o f diameter two and some r e l a t e d s t r u c t u r e s ( t o appear).
[ 1 7 1 B. Segre, L e c t u r e s on modern geometry. With an Appendix by L. Lombardo-Rad i c e , Cremonese, Roma (1961).
[ 1 8 ] J.C. Stempe, 266-280.
Geodetic graphs o f diameter two, J. Comb. Theory
B 17 (1974)
143
F-BINOMIAL COEFFICIENTS AND RELATED COMBINATORIAL TOPICS: PERFECT MATROID DESIGNS, POSETS OF FULL BINOMIAL TYPE AND F-GEODETIC GRAPHS P i e r V i t t o r i o C e c c h e r i n i and Anna Sappa D i p a r t i m e n t o d i Matematica "G. Castelnuovo" Uni v e r s i t l d i Roma "La Sapi enza" C i t t a U n i v e r s i t a r i a , 00100 Roma, I t a l y
We i n t r o d u c e F-binomial c o e f f i c i e n t s as a n a t u r a l g e n e r a l i z a t i o n o f b i n o m i a l and q - b i n o m i a l c o e f f i c i e n t s . A g e n e r a l c a l c u l u s w i t h these numbers l e a d s t o u n i f y t h e a r i t h m e t i c a l p r o p e r t i e s o f ( f i n i t e ) p r o j e c t i v e and a f f i n e spaces and o f S t e i n e r systems S(t,k,v) i n t o those o f p e r f e c t matroid designs ( 5 2 ) . P a r t i a l l y o r d e r e d s e t o f f u l l b i n o m i a l t y p e ( 5 3 ) and graphs such t h a t t h e number o f geodesics between any two v e r t i c e s depends o n l y on t h e i r d i s t a n c e ( 5 4 ) a r e a l s o s t u d i e d by means o f t h i s f o r m a l c a l c u l u s .
1. F-BINOMIAL COEFFICIENTS Let N (resp.
Q ) denote t h e s e t o f non n e g a t i v e i n t e g e r s ( r e s p . r a t i o n a l
numbers) and l e t be N* = N \ ( O I , L e t F: N
+
Q*
=
Q\rO).
Q* be any f u n c t i o n such t h a t F ( 0 )
any f u n c t i o n such t h a t f ( 0 ) = 0, f ( 1 )
=
1 and f ( N * )
=
F ( 1 ) = 1 and l e t f : N
+
Q be
5 Q*.
F and f w i l l be f u n c t i o n s s a t i s f y i n g t h e above c o n d i t i o n s .
I n what f o l l o w s ,
Given an F, t h e a s s o c i a t e d f w i l l be d e f i n e d by: f(0)
0, f ( n )
F(n)/F(n-l) for n 1 1 .
=
Given an f, t h e a s s o c i a t e d F w i l l be d e f i n e d by: F ( 0 ) = 1, F ( n ) = f ( n ) F ( n - 1 )
i . e . F(n) = f ( n ) f ( n - l ) . . , f ( l )
Two such f u n c t i o n s F and f a r e t h e n m u t u a l l y associated,
f o r n 31.
and sometimes below t h e y
w i l l be used i n t e r c h a n g e a b l y . DEFINITION 1.1.
Given a p a i r ( F , f ) o f m u t u a l l y a s s o c i a t e d f u n c t i o n s and g i v e n
any i n t e g e r s k,n w i t h
OC
n
k < n , we s h a l l d e f i n e t h e F-binomial c o e f f i c i e n t ( o r
b i n o m i a l c o e f f i c i e n t ) I 1 as t h e r a t i o n a l number k F
f-
144
P. V. Ceccherini and A. Sappa
These numbers t u r n o u t t o be p o s i t i v e i n t e g e r s i n t h e f o l l o w i n g examples ( i n which f and F a l s o t a k e i n t e g e r v a l u e s ) .
EXAMPLE 1.2.
Binomial c o e f f i c i e n t s . Consider t h e p a i r o f m u t u a l l y a s s o c i -
ated functions
fl(t)
=
and
t
F ( t ) = t! 1
for a l l tEN.
Then n
in)= k F1
= Ik} i s t h e usual b i n o m i a l c o e f f i c i e n t
k fl
(O
EXAMPLE 1.3. Gaussian numbers. Given an i n t e g e r q > 1 , c o n s i d e r t h e p a i r o f m u t u a l l y associated f u n c t i o n s
f (t) = [ t l 9 9 where [ ]
9
[Ol,!
F ( t ) = [t],! 9
and [ ] ! a r e d e f i n e d by q
9
[OI
and
=
[ll
0,
=
=
9
[ll ! q
[tlq
1,
= q
t-1
t
... + q t l ,
[ t l q l= [ t l [ t - 1 1
= 1,
q
q
... [ llq,
t 22.
Then
n - n I k I F - ikIf 9 q
n
[,Iq i s
the q-binomial coefficient
EXAMPLE 1.4. Constant c o e f f i c i e n t s .
( g a u s s i a n number, c f . 141).
Given an i n t e g e r a z 1, c o n s i d e r t h e
p a i r o f mutually associated functions f(0) = 0,
f(1)
=
1,
f ( t ) = a;
F ( 0 ) = F ( 1 ) = 1,
F ( t ) = at - ’
(tz2).
Then n n I 1 = i l =1, n F OF
and
n
IkjF =
a
for
Usual b i n o m i a l i d e n t i t i e s and q-binomial
O < k
are p a r t i c u -
I45
F-Binomial Coefficients and Related Combinatorial Topics l a r cases o f F-binomial i d e n t i t i e s ( o b t a i n e d f o r F
F
=
1 g i v e now some o f them, w r i t t e n a s f - b i n o m i a l i d e n t i t i e s .
and f o r F
=
F r e s p . ) . We q
P R O P O S I T I O N 1.5. The f o l l o w i n g f - b i n o m i a l i d e n t i t i e s h o l d :
n ‘k’f
- f(n) - fo
n
-
Ikff
-
n-1
I k
{
If +
n-1 k If
-
f(n)
fo
f(n)-f(n-k) f(k)
n-1 ‘k-l’f
- f(n-k+l) n f(k) ‘k-l’f
(O< k tn),
n-1 (k-l’f
These f o r m u l a s suggest t h e f o l l o w i n g DEFINITION 1.6. Given a f u n c t i o n f and any i n t e g e r s k,n w i t h 0 < k
PROPOSITIONS 1.7. L e t
equivalent: (a)
dfn,k
(c)
be as Def. 1.6. The f o l l o w i n g c o n d i t i o n s a r e
i s independent o f n;
n,k
(b) f = f
n,k
9
f o r some q E Q * (where f
= qk
f o r some
f (n)-f (k) 9 9 f q(n-k) bf
n,l
f(n) =
=
=
9
n-1
+ . . . +q
..+q+l
qn-k-1,.
i s f o r m a l l y d e f i n e d as i n Ex. 1 . 3 ) .
q E Q*.
P r o o f . O b v i o u s l y ( c ) = + ( a ) . We have ( b ) -
q
f 9
( c ) because
=
gf
n,k
A
n,k
k
(b).
= qk. We prove now t h a t ( a )
=
L e t us p u t
q f o r a l l n, Then f ( 1 ) = 1 and f o r a l l i n t e g e r s n 3 2 we g e t Af
n,l
f(n-l)+f(l),
so t h a t t h e e q u a l i t y f ( n )
=
q
n-1
+
...
+ q+l f o l l o w s by
induction.
2 . PERFECT M A T R O I O DESIGNS A c o m b i n a t o r i a l geometry ( o r s i m p l e m a t r o i d ) M on a f i n i t e s e t S i s a m a t r o i d
P.V. Ceccherini and A. Sappa
146
0, S andevery s i n g l e t o n o f S a r e f l a t s o f M ( c f . [ 2 1 , [911. A p e r f e c t m a t r o i d d e s i g n (PMD) i s a c o m b i n a t o r i a l geometry M such t h a t e v e r y k - f l a t ( f l a t of rank k ) has t h e same number f ( k ) o f p o i n t s , k = O,l, ...,r k M
on S such t h a t
[lo],
(cf. f(0)
[ 1 2 ] , [ 1 4 1 ) . We s h a l l say t h a t f i s t h e s i z e - f u n c t i o n o f M. Obviously
n
0 and f ( 1 ) = 1, so t h a t f - b i n o m i a l c o e f f i c i e n t s t k l f can be
=
considered
(Osk(n4r-k MI. X PMDs i n c l u d e boolean s e t s 2 , p r o j e c t i v e spaces, a f f i n e spaces, t - ( v , k , l )
designs ( c f . [ 1 4 ] ) . PROPOSITION 2.1.
L e t M be a PMD on a f i n i t e s e t S w i t h s i z e - f u n c t i o n f .
I n t h i s case ( a ) M i s t h e m a t r o i d o f a l l subsets o f S i f and o n l y i f f = f 1' n n f f ( n ) = n, 1 ~ = 1( k ) ~ and = 1.
( b ) M i s t h e m a t r o i d o f f l a t s o f a p r o j e c t i v e space o f o r d e r q i f and o n l y n n-1 n i f f = f ( w i t h q a 2 ) . I n t h i s case f ( n ) = q t ... t q t l , 1 ~ = 1[ 1 ~ and f ",k
9
k q
(Conversely, each o f t h e s e e q u a l i t i e s i m p l i e s f = f 1. 9 n-1. ( c ) IfM i s thematroid o f f l a t s o f an a f f i n e space o f o r d e r q, t h e n f ( n ) = q , n f k 2k-n t h e converse i s t r u e when q a 4. I n t h i s case { k ] f = qk(n-kl and = q - q . n-1 1. (Conversely, each o f these e q u a l i t i e s i m p l i e s f ( n ) = q
q
=
*
P r o o f . ( a ) i s obvious. For ( b ) ( r e s p . ( c ) ) , we have t o prove o n l y t h e " i f " p a r t . A s i m p l e c o u n t i n g argument shows t h a t e v e r y 3 - f l a t i s a p r o j e c t i v e ( r e s p . an a f f i n e ) p l a n e o f o r d e r q, so t h a t t h e r e s u l t f o l l o w s f r o m a w e l l known charact e r i z a t i o n o f p r o j e c t i v e ( r e s p . a f f i n e ) spaces by means o f planes, c f . [ Z ] ( r e s p .
[ll).
0
With an argument which i s s t a n d a r d f o r f i n i t e p r o j e c t i v e spaces ( c f . [14] one can prove t h e f o l l o w i n g PROPOSITION 2.2.
L e t M be a PMD w i t h s i z e - f u n c t i o n f.
( a ) The number o f k - f l a t s i n c l u d e d i n a r - f l a t ( w i t h O < k 6 r ) i s g i v e n by
- 'r, 1 " ' ' r , k - l A k, 1 * k, k-1
'
...f ( r - k + l )
f(r)
.
f(k). .f(l)
--
A ~ , ~ . . . A ~ , ~ - ~
'k, 1
"
'Ak,k-l
'k'f'
( b ) When a k - f l a t i s i n c l u d e d i n a r - f l a t , t h e number o f j - f l a t s between them
(01:k < j c r ) i s g i v e n by
147
F-Binomial Coefficients and Related Cornbinatorial Topics
I n p a r t i c u l a r B(O,j,r) B(k,ktl,r)
= cr(j,r)
and
"" f ( r - k ) A
=
k+l, k
( c ) When a k - f l a t A i s i n c l u d e d i n a r - f l a t 6, t h e number o f maximal c h a i n s
s g i v e n by
o f f l a t s between them ;(k,r)
...B ( r - 2 , r - l , r )
r ) B(ktl,kt2,r)
B(k,ktl
=
=
Ar,k.'.Ar,r-2 Ak
t l ,k ' * ' A r -,lr - 2
F(r-k)
where F i s a s s o c i a t e d t o f . L e t M be a PMD on a f i n i t e s e t S w i t h s i z e - f u n c t i o n f (and
PROPOSITIONS 2.3.
a s s o c i a t e d f u n c t i o n F ) and l e t a ( k , r ) , t i o n 2.2.
B(k,j,r),
;(k,r)
be d e f i n e d as i n p r o p o s-i
The f o l l o w i n g c o n d i t i o n s a r e e q u i v a l e n t :
( 1 ) M i s t h e PMD o f a g r a p h i c space S o f o r d e r q ( i . e . a p r o j e c t i v e space o f o r d e r q ( p o s s i b l y a " l i n e " ) , when 9 2 2 , o r t h e boolean s e t PS, when q
r-k =(j-k~
(lb) B(k,j,r)
( l b ' ) B(k,k+l,r) ( l c ) ;(k,r)
=
Moreover, i f c o n d i t i o n ( 1 (2) f
=
f
q'
for all
= f r-k)
F(r-k)
a(k,r)
D < k.S r
for all
( l a ) a ( k , r ) = {;If
1);
c r k M;
0 6 k < ;i < r s r k M;
for all
for a l l
=
O s k i r s r k M;
O < k < r < r k M.
i s s a t i s f i e d , then r k q'
= [ ]
l3(k,j,r)
=
[;][I,,
;(k,r)
= [r-kl!
P r o o f . I t i s w e l l known ( c f . [ 3 ] , [ 1 6 ] ) t h a t ( 1 ) i m p l i e s ( l a ) - ( l c ) and ( 2 ) . (la)-(l).
By Prop. 2.2,
we g e t
Ar,l r r a ( 2 , r ) = A* {21f -- 121f
231
s
P.V. Ceccherini and A. Sappa
148
(Ib')
=,
( 1 ) . By Prop. 2.2,
we g e t R ( k , k t l , r )
=
'r k 'ktl
4
r,k
=
'k+l,k'
.
f o r k = l we o b t a i n a(k,r)
A
= A
r,l
= R(O,k,rl
f(r-k)
= f(r-k)
and ( 1 ) f o l l o w s as above.
2,l
r-0 I k-O'f
=
,k
r - 'k'f' -
(lb)
=,
(la).
(Ic)
=)
( l b ) . I n each maximal c h a i n of f l a t s between a k - f l a t A and a r - f l a t
B t h e r e e x i s t s e x a c t l y one j - f l a t C ( 0 k ~c j < r ( r k M ) ; g i v e n such a C, t h e j o i n -
i n g of a maximal c h a i n between A and C and o f a maximal c h a i n between C and El i s a maximal c h a i n between A and B. {(C,$):C
between A and B), g i v e s :
i s a j - f l a t belonging t o a chain 0
F(r-k)
Nk,j,r)
=
REMARK 2.4. define
The double c o u n t i n g argument, a p p l i e d t o t h e s e t
a(k,r),
F(j-k)F(r-j),
B(k,j,r)
- {r-k j-kff.
F(r-k)
= F(r-jlF(j-kl
0
L e t M be t h e m a t r o i d o f f l a t s o f an a f f i n e space o f o r d e r q and R(k,j,r)
+
and v ( k , r ) as i n Prop. 2.2. From ( 2 ) o f Prop. 2.3,
f o l l o w s t h a t f ( 0 ) = 0, f ( t ) = qt-', r-k a(k,r) = q REMARK 2.5. such t h a t
i.e.
(k,r)
F ( t ) = qt(t-1)'2
r-1
[k-llq, B ( k , j , r )
=
[;:;Iq,
;(k,r)
it
and f o r k > O = [r-kl
q
!.
We g i v e now an o t h e r example o f a PMD M with s i z e - f u n c t i o n f
# F ( r - k ) . L e t S be a S t e i n e r system S(Z,k,n)
(e.g. t h e S t e i n e r
system whose b l o c k s a r e t h e l i n e s o f a p r o j e c t i v e ( r e s p . a f f i n e ) space S o f o r d e r q w i t h k = q t l ( r e s p . k = q ) ) and l e t M be t h e PMD ( o f r a n k 3 ) whose f l a t s a r e t h e empty s e t ,
the s i n g l e points,
t h e b l o c k s and t h e f u l l s e t o f p o i n t s . The s i z e
f u n c t i o n f i s such t h a t f ( 0 ) = 0, f ( 1 ) = 1, f ( 2 ) = k, f ( 3 ) = n; t h e a s s o c i a t e d f u n c t i o n F i s such t h a t F(0) = F ( 1 ) = 1, F ( 2 ) = k, F ( 3 ) = nk; moreover =
= 1,
A
2,l
= k-1,
A
3,l
=-
'-' . k
=
Therefore
2 whenever n f k - k + l ( i n t h e p r e v i o u s examples, whenever t h e space S i s n o t a p r o j e c t i v e plane).
149
F-Binomial Coefficients and Related Combinatorial Topics 3. POSETS OF FULL
BINOMIAL TYPE
These s t r u c t u r e s have been i n t r o d u c e d i n [111, ( c f . a l s o [71). We s h a l l use t h e n o t a t i o n o f $1.
(P,*.) be a p a r t i a l l y o r d e r e d s e t ( p o s e t ) w i t h minimum 0, and l e t a,b be elements o f P. I f a 4 b, t h e i n t e r v a l between them i s d e f i n e d by I ( a , b ) = Let
=
f c ~ P :a < c c b l . I f a < b a maximal c h a i n between a and b i s a c h a i n
a = a < a <...< an = b o f I ( a , b ) which i s n o t i n c l u d e d i n a l o n g e r c h a i n o f 0 1 I ( a , b ) ; t h e number n i s c a l l e d t h e l e n g t h o f t h e c h a i n . We s h a l l denote by ;(a,b) t h e number o f maximal c h a i n s o f I ( a , b ) , We s h a l l say t h a t (JD)
and we s h a l l assume
;(a,b)
=
1 when a=b.
P i s a JD-poset i f i t s a t i s f i e s t h e f o l l o w i n g c o n d i t i o n
(Jordan-Dedeking c h a i n c o n d i t i o n ) : i f a,b
E
f
w i t h a < b, a l l t h e maximal
c h a i n s o f I ( a , b ) have one and t h e same l e n g t h , denoted by d(a,b)
and c a l l e d t h e
d i s t a n c e between a and b ( o r t h e l e n g t h o f t h e i n t e r v a l ( I ( a , b ) ) . If a=b, we p u t d(a,b)
=
If
0.
P i s a JD-poset w i t h minimum 0, t h e n t h e rank o f an element a
d e f i n e d by r k a=d(O,a), and t h e rank o f We n o t e t h a t i f a,bEP
there exists a function F : f O , l , ;(a,b)
interval o f length n o f that
i s d e f i n e d by r k
w i t h a < b, t h e n d(a,b)<
We s h a l l say t h a t a JD-poset a < b, t h e number
P
rk
P
P
P.
P ( w i t h minimum 0 ) has a c h a i n f u n c t i o n if
...,r k
PI
+
N such t h a t , f o r a l l a,b E 2 w i t h i.e.
every
has F ( n ) maximal c h a i n s . I n t h i s case we s h a l l say
p i s an F-chain p o s e t . A poset
DEFINITION 3.1. (1)
P
i s c a l l e d a poset o f f u l l b i n o m i a l t y p e i f
P has minimum 0,
(2) P
i s a JD-poset,
(3)
P i s a F - c h a i n poset, f o r some F.
If
f i s a JD-poset w i t h minimum 0, f o r each i n t e r v a l I ( a , b )
each i n t e g e r k, w i t h O s k s d ( a , b ) , Ik(a,b)
= I c E I(a,b)
We s h a l l say t h a t f : IO,1,
...,r k
+ .
P and f o r
of
we s h a l l p u t
: d(a,c) = k)
P has a s i z e f u n c t i o n _ i f t h e r e e x i s t s a f u c n t i o n
P)
is
= max t r k a: a € $ , .
o f maximal c h a i n s between them i s F ( d ( a , b ) ) ,
P
E
N such t h a t , f o r a l l i n t e r v a l s I ( a , b ) o f f ,
f,
I50
P.V. Ceccherini and A . Sappa IIl(a,b)I
= f(d(a,b)).
I n t h i s case we s h a l l say t h a t
P
i s an f - s i z e poset. The f o l l o w i n g p r o p o s i -
t i o n y i e l d s o t h e r e q u i v a l e n t d e f i n i t i o n s o f poset o f f u l l b i n o m i a l t y p e . PROPOSITION 3.2.
P
Let
be a JD-poset w i t h minimum 0; t h e f o l l o w i n g c o n d i -
t i o n s a r e e q u i v a l e n t , where f and F a r e m u t u a l l y a s s o c i a t e d f u n c t i o n s : (a)
( b l IIk(a,b)I (c)
P
P i s an F-chain poset ( i . e .
i s a poset o f f u l l binomial type).
(Oc k c d(a,b)).
= t d(a,b)]
P i s an f - s i z e poset.
Proof. Let a , b E P , (a) * ( b ) . e x a c t l y one c E
acb.
L e t be O < k < d ( a , b ) .
I n each maximal c h a i n o f I ( a , b ) t h e r e e x i s t s
P such t h a t d(a,c)
= k.
I f c i s such an element o f I ( a , b ) ,
the
j o i n i n g o f a maximal c h a i n o f I ( a , c ) and o f a maximal c h a i n o f I ( c , b ) i s a maximal c h a i n o f I ( a , b ) . CL
The double c o u n t i n g argument a p p l i e d t o t h e s e t I ( c , o ) :
i s a maximal c h a i n o f I ( a , b ) , F(d(a,b)) = IIk(a,b)
I
d(a,c)
= k)
C E ~ ,
gives:
F ( k ) F(d(a,b)-k),
so t h a t ( b ) f o l l o w s .
( b ) * ( c ) . For k = l we o b t a i n I1 ( a , b ) I = I d(;yb))f = f ( d ( a , b ) . 1 ( c ) * ( a ) . A c t u a l l y I 1 ( a , b ) I = f ( d ( a , b ) ) . We now a p p l y i n d u c t i o n on d ( a , b ) ; 1 i f d(a,b) = 0,1, t h e n 1 = ;(a,b) = F ( d ( a , b ) ) . I f d(a,b),2, a p p l y t h e double coun t i n g argument t o t h e s e t { ( c , ~ ) : c E
U,U
i s a maximal c h a i n o f I ( a , b ) ,
d(a,c)=ll.
By t h e p r e v i o u s argument and b y t h e i n d u c t i o n h y p o t h e s i s , we have ;(a,bl
= (Il(a,b)I
EXAMPLE 3.3.
F(1) F(d(a,bl-l)
= f(d(a,b))
P= 2
I f X i s a f i n i t e s e t and i f
o f X o r d e r e d by i n c l u s i o n , t h e n
P
F(d(a,b)-l)
X
= F(d(a,bl).
i s the set o f a l l
0
subsets
i s a poset o f f u l l b i n o m i a l type, w i t h s i z e
f u n c t i o n f ( t ) = t and c h a i n f u n c t i o n F ( t ) = t ! ( 0 t ~c
1x1)
any such f u n c t i o n s f and F ( w i t h dom f = dom F = t O , l ,
...,n l ) ,
(cf.
[ 1 3 1 ) . Conversely
( c f . example 1.21, X can be c o n s i d e r e d as s i z e f u n c t i o n and c h a i n f u n c t i o n o f a boolean s e t p = 2 w i t h 1x1 = n. We n o t e t h a t , i f P = I X I U ( ox ) U ( ,x) U...U ( Xk ) w i t h 2 4 k < 1x1 - 1, t h e n t h e
P i s n o t of f u l l b i n o m i a l t y p e : i f Y
poset
= d(Z,X)
z
to
x.
= 2, b u t t h e r e a r e
X
E ( 2 ) and
2 c h a i n s from 0 t o Y and
Z E(krl)l
1x1
-
t h e n d(0,Y)
=
(k-1) > 2 chains from
F-Binomial Coefficients and Related Combinatorial Topics EXAMPLE 3.4. dimension n-1
IS1
If X i s a f i n i t e p r o j e c t i v e space PG(n-1.q)
o f o r d e r q and
2 and i f ?' i s t h e s e t o f f l a t s o f X o r d e r e d by i n c l u s i o n , t h e n ?
i s a poset o f f u l l b i n o m i a l t y p e w i t h s i z e f u n c t i o n f ( t ) = [ t ]
and c h a i n f u n c t i o n q Conversely, any such f u n c t i o n s f = [ ] and
F ( t ) = [ t ] !. ( c f . [ 3 ] , prop. 3.3.). 9 h F = [ I ! w i t h q = p ( h h l , p p r i m e 3 2 1 and dom f = d a n F = 10.1
,...,n l
f-l
q
( c f . exaE
p l e 1 . 3 ) can be c o n s i d e r e d as t h e s i z e and t h e c h a i n f u n c t i o n s o f t h e poset o f t h e subspaces of a p r o j e c t i v e space X = PG(n-l,q) We n o t e t h a t , if with i
=
P
o f o r d e r q and dimension n-1.
i s t h e s e t of a l l t h e i - d i m e n s i o n a l f l a t s o f X = PG(n-l,q),
-l,O,...,k,n-1
where 1 4 k < n - 2 , t h e n t h e poset
P i s not o f f u l l binomial
t y p e : i f Y i s a l i n e and 2 i s a ( k - 1 ) - d i m e n s i o n a l f l a t , t h e n d(0,Y) b u t t h e r e a r e q + l c h a i n s f r o m 0 t o Y and q wk-' +
EXAMPLE 3.5. I f
$=
...+q+l > q + l
,,..., xla ,..., xnl ,...,x na 1
{xo,xl
elements (a,n a l ) , o r d e r e d by assuming xo as minimum and x
= d(Z,X)
= 2
c h a i n s f r o m Z t o X.
i s any s e t w i t h a n t 1 ij
< x
hk
i f f i < h, t h e n
2 i s a p o s e t o f f u l l b i n o m i a l t y p e w i t h s i z e f u n c t i o n f ( O ) = O , f ( l ) = l , f ( t ) ad with t-1 chain functionF(O)=F(l)=l,F(t) = a ( 2 6 t < n ) . Conversely, any such f u n c t i o n s f and F ( c f . example 1.3.) can be c o n s i d e r e d as t h e s i z e and t h e c h a i n f u n c t i o n s o f a poset o f f u l l b i n o m i a l t y p e p as above. REMARK 3.6.
L e t M be t h e PMD o f t h e f l a t s o f an a f f i n e space X = AG(n-1,q) and l e t B
o f dimension n - 1 3 2 ,
not o f f u l l binomial type: t h e n d(0,Y)
z
to
= d(Z,X)
be t h e p o s e t o f t h e f l a t s o f X. The poset
P
is
i f Y i s a l i n e and Z i s an ( n - 3 ) - d i m e n s i o n a l f l a t ,
= 2 , b u t t h e r e a r e q c h a i n s f r o m 0 t o Y and q + l c h a i n s f r o m
x. Therefore,
if
M i s a PMD on a f i n i t e s e t X and if P i s t h e s e t of t h e f l a t s
o f M ordered by inclusion, then
P i s not necessarily a poset o f f u l l binomial
t y p e : t h e number o f maximal c h a i n s i n an i n t e r v a l I ( a , b ) does n o t depend o n l y on t h e l e n g t h o f t h e i n t e r v a l , b u t a l s o on t h e r a n k s o f a and b ( c f . a l s o Remark 2.5,
and Examples 3.3,
3.4).
The case when
P i s a poset o f f u l l b i n o m i a l t y p e
i s characterized by t h e f o l l o w i n g proposition. PROPOSITION 3.7.
L e t M be a PMD on a f i n i t e s e t X and l e t
t h e f l a t s o f M o r d e r e d b y i n c l u s i o n . The poset
P
P
be t h e poset o f
i s of f u l l b i n o m i a l t y p e i f and
o n l y i f M i s one o f t h e f o l l o w i n g PMD's: ( a ) M i s t h e t r i v i a l PMD o f r a n k 1 o r t h e t r i v i a l PMD o f r a n k 2 on X ( i . e .
M i s a t r i v i a l g r a p h i c space o f dimension 0 on 1 on X r e s p . ) ;
152
P.V. Ceccherini and A . Sappa (b) M i s t h e m a t r o i d 2
X
o f a l l subsets o f X, i . e . M i s t h e g r a p h i c space o f
o r d e r 1 and dimension 1x1-1;
( X I o f a l l f l a t s o f a p r o j e c t i v e space, o f dimenY q h a v i n g X as s e t o f p o i n t s .
( c ) M i s t h e m a t r o i d Mn-l s i o n n-1 and o r d e r q * 2 ,
P r o o f . I f M i s a PMD as i n ( a ) - ( c ) , t h e n t h e poset
P
o f i t s f l a t s i s a poset
o f f u l l b i n o m i a l t y p e ( c f . Ex. 3.3 and Ex. 3.4). L e t M b e a PMD on X such t h a t t h e p o s e t b i n o m i a l type. I f r k
M r 2,
B o f i t s f l a t s i s a poset o f f u l l
t h e n we a r e i n t h e case ( a ) . If r k
M > 2,
l e t f denote
t h e s i z e f u n c t i o n o f M. F o r any f l a t Y o f r a n k 2, we have t h a t lIl(O,Y)l
P
I n o t h e r words, t h e p o s e t
= ( Y I = f ( 2 ) = f(d(0,Y)).
has t h e same s i z e f u n c t i o n f t h a n t h e PMD M. Thus
c o n d i t i o n ( b ) o f Prop. 3.2 holds; i t means t h a t c o n d i t i o n ( l b ) o f Prop. 2.3 h o l d s . So c o n d i t i o n ( 1 ) o f Prop. 2.3 h o l d s too, and t h e r e s u l t ( b ) - ( c ) i s proved.
0
4. F-GEODETIC GRAPHS a l l graphs w i l l be f i n i t e w i t h o u t l o o p s o r m u l t i p l e edges,
I n what f o l l o w s ,
an:i a l l d i r e c t e d graphs w i l l be w i t h o u t d i r e c t e d c i r c u i t s . Any d i r e c t e d graph
p =
p(t) =
G'
(V,;)
=
i s o b v i o u s l y t h e Hasse diagram o f a poset
where x < y i f and o n l y if t h e r e e x i s t s a d i r e c t e d p a t h f r o m x
(V,<)
t o y; c o n v e r s e l y t h e Hasse diagram o f a poset p = (V,<)
6
= (V,:),
=
that
E
where ( x , y )
P(E) a r e
and
I f G = (V,E)
EE
i s a d i r e c t e d graph
i f and o n l y i f x i s covered by y. We s h a l l say
mutually associated.
(resp.
6
*
= (V,E))
i s a graph ( r e s p . d i r e c t e d graph) and i f two
v e r t i c e s x,yE V a r e j o i n e d by a p a t h ( r e s p . d i r e c t e d p a t h ) , t h e d i s t a n c e d ( x , y ) ( r e s p . i ( x , y ) ) i s d e f i n e d as t h e number o f edges i n a geodesic i . e . p a t h (resp. d i r e c t e d s h o r t e s t p a t h ) between x and y. We denote by (resp. b y
*
r (x,y)
+
= rG(x,y))
i n a shortest r(x,y) =rG(x,y)
t h e s e t o f d i s t i n c t geodesics o f G (resp.
of
between x and y; we p u t : v(x,y) q(x,x)
=
I
r(x,y)l
y(x,x)
= 1
diam G = max I d ( x , y )
and ;(x,y) and
=
;I
x,y)I +
d ( x , x ) = d x,x)
: x,y E
Vj,
d am
We s h a l l say t h a t a connected graph G =
E
V,E)
ifx # y; = 0;
= max ~+d(x,y) : x,y E W .
( r e s p . a d i r e c t e d graph
5
= (V,:))
El
153
F-Binomial Coefficients and Related Combinatorial Topics
...,diam
i s a graph w i t h a geodetic f u n c t i o n i f t h e r e e x i s t s a map F : ( O , l , (resp. F : ( O , l ,
*
...,diam
G)
N ) such t h a t u ( x , y ) = F ( d ( x , y ) ) ( r e s p .
+
+
GI
+
N
u(x,y) =
i) i s
= F(J(x,y))
f o r a l l x,y E V .
F-geodetic.
Note t h a t F(0) = 1 and t h a t F(n) # 0 f o r a l l n E dom F, so t h a t t h e as -
I n t h i s case we s h a l l a l s o say t h a t G (resp.
sociated f u n c t i o n f can be considered as i n $1. Let
2
= (V,;)
be a d i r e c t e d graph and l e t x , y ~ V be such t h a t x - < y . Then t h e
i n t e r v a l I ( x , y ) i s defined by I(x,y) := I z E V :
+
x < z - ( y l , i . e . I ( x , y ) i s defined as i n P ( G ) ,
and t h e geodetic i n t e r v a l I g ( x , y ) i s d e f i n e d by P(x,y)
:= t z e V:
Note t h a t I 9 (x,y)
C;(x,y) f o r some
ZE
5 I(x,y)
C; E r ( x , y ) ~ .
and t h a t I g (x,y) = t z e I ( x , y ) : i ( x , z )
= ;(x,y)-;(y,z)).
For any 1 < k < i ( x , y ) , l e t : = I z e I g ( x , y ) : ~ ( x , z )= k l
I;(x,y)
E
We say t h a t
= (V,:)
has a source
= (zE
I9 ( x , y ) : J ( Z , Y ) = i ( x , y ) - k l .
O E V i f f o r any
XE
V \ (01 t h e r e e x i s t s a d i r e c -
t e d path from 0 t o x.
A graph
6
= (V,i)
w i l l be c a l l e d a d i r e c t e d graph o f f u l l binomial t y p e ( w i t h
geodetic f u n c t i o n F ) i f : (a)
E
has a source 0,
( b ) f o r a l l x , y ~ Vw i t h x < y : (c)
Let
E
be t h e poset associated w i t h
(2)
= I(x,y),
G i s F-geodetic f o r some F.
PROPOSITION 4.1.
(1)
I 9 (x,y)
P has a minimum P i s a JO-poset
= (V,i)
6
be a d i r e c t e d graph and l e t P = P ( 6 ) = ( V , < )
(so t h a t
=
0 i f and o n l y i f
6(P)).
Then
has source 0;
i f and o n l y i f I g ( x , y )
I(x,y),
=
for a l l x , y ~ V
with x
6
P
i s a poset o f f u l l binomial type w i t h chain f u n c t i o n F i f and o n l y i f
i s a d i r e c t e d graph o f f u l l binomial type w i t h geodetic f u n c t i o n F. Proof. -
( 1 ) i s obvious. ( 2 ) :
s e t M(x,y) o f maximal chains o f G(x,y;of
E
P i s a JD-poset P i n I ( x , y ) i s the
O f o r a l l x , y ~ Vw i t h x c y
0
f o r a l l x , y ~ Vw i t h x < y t h e set
I(x,y) = Ig(x,y).
? ( x , y ) o f t h e geodesics
( 3 ) : P i s a poset
of
P.V . Ceccherini and A . Sappa
154
f u l l binomial type w i t h chain f u n c t i o n F
I
(M(x,y) Ig(x,y)
F(+d(x,y)) f o r a l l x < y i n V
=
= I(x,y)
and [;(x,y)(
= F(d(x,y))
t y p e w i t h g e o d e t i c f u n c t i o n F. The poset
p = (V,<),
o
O P has minimum 0, I' i s a JD-poset and
"G
has source 0, f o r a l l x < y i n V
E
i s a d i r e c t e d graph o f f u l l b i n o m i a l
0
where V = IO,x,y,z,tl
and O < x < z < y , O < t < y , i s n o t a
b u t t h e graph 6 ( p ) i s F-geodetic ( w i t h F = l ) ; n o t e t h a t Ig(O,y)
JD-poset, = {O,t,Yl
c I(0,y)
=
=
v.
PROPOSITION 4.2.
Let
"G
=
be a d i r e c t e d graph. The f o l l o w i n g c o n d i t i o n s
(V,i)
a r e e q u i v a l e n t (where F and f a r e m u t u a l l y a s s o c i a t e d ) : i s F-geodetic f o r some F,
(a)
( b ) f o r a l l x < y i n V and f o r a l l 1 g k c a ( x , y ) : (c) f o r a l l x < y i n V:
II~I=
*
(c)
=)
=
+
1
t o t h e s e t Z1 g i v e s : = F(a(~,y)).
If
2
ated t o
E.
gives:
F.
1.
( a ) . We a p p l y i n d u c t i o n on a ( x , y ) 3 1 . I f a ( x , y )
= F(i(z,y))
z(x,z) = k l
+
9 d ( x ,y so t h a t I k ( x , y ) = I
= 1 = F ( 1 ) . Suppose d ( x , y ) h 2 ; by t h e i n d u c t i o n hypothesis,
IT(z,y)
J(X,Y) k IF'
The double c o u n t i n g argument
(z,<(x$y)): z ~ < ( x , y ) ,ger(x,y),
=
F ( d ( x , y l ) = I I z ( x , y ) l F ( k ) F(a(x,y)-k), (b) * (c) for k
k
f(i(x,y)).
Proof, L e t x,y be elements o f V w i t h x < y . ( a ) - ( b ) . a p p l i e d t o t h e s e t Zk
9
( I (x,y)I = i
1
I
=
we have 1 Therefore t h e double c o u n t i n g argument a p p l i e d
= F(G(x,y)-l).
I?(x,y)
1, t h e n I;(x,y) when Z E Z
= IIg(x,y) l F ( l ) F ( i i ( ~ , y ) - l ) = f ( a ( x , y ) )
1
F(a(x,y)-l)=
0
i s a d i r e c t e d graph, we s h a l l denote by G t h e u n d i r e c t e d graph a s s o c i -
PROPOSITION 4.3.
Let
= (V,:)
be a d i r e c t e d graph w i t h source O S V . Suppose
t h a t f o r each x E V t h e p a r i t y o f t h e l e n g t h o f any d i r e c t e d path f r o m 0 t o x depends o n l y on x; w r i t e p ( x ) =O i f i t i s even and p ( x ) = 1 i f i t i s odd. Then +
t h e u n d i r e c t e d graph G a s s o c i a t e d t o G i s connected and b i p a r t i t e . P r o o f . G i s connected s i n c e 0 i s a source. We v e r i f y t h a t G i s b i p a r t i t e by 0 1 i assuming V = V U V where V = I x E V : p ( x ) = il, i = 0 , l . We have t o prove t h a t i f ( x , ~ ) EE then p ( x ) # p ( y ) . We can suppose w i t h o u t loss o f g e n e r a l i t y t h a t (X,Y)E
E.
I f G(0,x) and G(0,y)
a r e geodesics f r o m 0 t o x and t o y resp.,
then
F-Binomial Coefficients and Related Combinatorial Topics
155
g(0,y) and G(0,x) u (x,y) are both d i r e c t e d paths from 0 t o y, so t h a t t h e i r lengths have t h e same p a r i t y p ( y ) . It f o l l o w s t h a t p ( x ) # p ( y ) . PROPOSITION 4.4.
L e t G = (V,E) be a connected b i p a r t i t e graph and l e t 0 be
any vertex o f G. Then by s t a r t i n g from 0, a n a t u r a l o r i e n t a t i o n can be d e f i n e d
on E, i n such a way t h a t
= (V,:)
i s a d i r e c t e d graph w i t h source 0 (and w i t h o u t
d i r e c t e d c i r c u i t s ) . It f o l l o w s t h a t
$=
where x
E
Proof. I f x,y E V w i t h d(0,x)
-*
p(G)
=
(V,C) i s a poset w i t h minimum 0,
a d i r e c t e d path from x t o y .
= d(O,y),
then x and y cannot be adjacent be-
cause G i s b i p a r t i t e . Let ( x , y ) be an edge o f G. We have e i t h e r d(0,y) = d(O,x)+l o r d(O,x)
= d(0,y)tl.
Orient t h e edge from x t o y i n the f i r s t case, from y t o +
x i n t h e second. It i s easy t o show t h a t , whenever t h e r e i s an edge (x,y) EE, then (x,y)
i s t h e o n l y d i r e c t e d path from x t o y, and t h a t whenever t h e r e e x i s t s
a d i r e c t e d path from x t o y then t h e r e i s no d i r e c t e d path from y t o x; indeed i s a d i r e c t e d path, then we have
(by i n d u c t i o n on i ) , i f (xo,x l,...,xi) (xo,xl)
,..., ( X ~ - ~1 , X . and ) E ~d(O,xi)
THEOREM 4.5.
= d(O,xo) t i.
0 +
+
L e t G = (V,E) be a connected b i p a r t i t e graph and l e t G = (V,E)
be the d i r e c t e d graph obtained by s t a r t i n g from a given vertex OEV as i n Prop.
4.4. Then, whenever x and y are elements o f
V w i t h x < y , the f o l l o w i n g c o n d i t i o n s
are e q u i v a l e n t : (a
p(x,y) i s a d i r e c t e d path from x t o y i n
(b
p(x,y)
(C
p(x,y) i s a geodesic from x t o y i n G.
Proof.
t,
i s a geodesic from x t o y i n 6, *
be any d i r e c a) * ( b ) . Assume x = 0 f i r s t . L e t p(0,y) = (O=x",,.,,x.=y) 1
t e d path o f lenght i from x t o y i n
5.
We have d(O,y)=i, by t h e i n d u c t i o n argu-
ment sketched a t the end o f t h e p r o o f o f Prop. 4.4.
Therefore p(0,y) = p ( x , y ) i s
a geodesic i n G. Assume now 0 < x < y . L e t p(0,x)
and p ( x , y ) be any d i r e c t e d paths i n
8
from
0 t o x and from x t o y resp. By g l u e i n g p(0,x) and p(x,y) we get a d i r e c t e d path p(0,y)
in
'G.
For t h e previous case p(0,y)
i s a geodesic i n G. Thus i t s subpath
p(x,y) must a l s o be a geodesic i n 6.
(b)
=)
( a ) . I n d u c t i o n on i = d ( x , y ) .
x < y . so t h a t p(x,y)
When i = l , we have p(x,y) = ( x , y ) ~ isince *
i s a d i r e c t e d path i n G. Assume now i Z 2 and suppose t h a t
P.V. Ceccherini and A . Sappa
156
any geodesic o f G o f l e n g t h j w i t h 1 6 j < i i s a d i r e c t e d p a t h i n
,...,x 1. - 1
p ( x , y ) = ( x =x,x
,xi=y)
and ( X ~ - ~ , Xo f~ G. ) These a r e b o t h d i r e c t e d p a t h s i n Then p ( x , y ) i s a d i r e c t e d p a t h i n (c) *(a)
obviously.
(b)*(c).
I f p(x,y)
6.
The geodesic
i s o b t a i n e d by g l u e i n g t h e geodesics p(x.xi-,)
6
by t h e i n d u c t i o n h y p o t h e s i s .
E.
...,x .1= y ) i s a geodesic i n G, t h e n p(x,y) i s 0 1' because (because ( b ) = . ( a ) ) , and i t must be a geodesic i n
a d i r e c t e d path i n
= ( x =x,x
6,
+
any s h o r t e r d i r e c t e d p a t h p ' ( x , y )
i n G would be a p a t h o f G s h o r t e r t h a n p ( x , y ) ,
which i s i m p o s s i b l e s i n c e p ( x , y ) i s a geodesic i n G. COROLLARY 4.6. L e t G
(V,E)
=
0
be t h e
be a connected b i p a r t i t e graph and l e t
d i r e c t e d graph o b t a i n e d by s t a r t i n g f r o m a g i v e n v e r t e x O E V as i n Prop. 4.4. Then ( 1 ) whenever x , y ~ V w i t h x r y , we have d ( x , y ) that
= +d(x,y), r ( x , y )
= ?(x,y)
so
r(x,y) = t(x,y); ( 2 ) G i s F-geodetic i f and o n l y i f
Proof.
E
i s F-geodetic f o r a l l O E V .
( 1 ) i s obvious. For ( 2 ) i t i s enough t o n o t e t h a t ;(x,y)
= r ( x , y ) when we
assume 0 = x. COROLLARY 4.7.
Let G
=
be a connected b i p a r t i t e graph. Then t h e f o l -
(V,E)
l o w i n g c o n d i t i o n s a r e e q u i v a l e n t , where F and f a r e m u t u a l l y a s s o c i a t e d f u n c t i o n s : ( a ) G i s F-geodetic,
IIzEV:
d(x,y)
( b ) x,yEV,
Otkcd(x,y)*
( c ) x,yEV
*
REMARK 4.8.
The statement o f C o r o l l a r y 4.7 also h o l d s i f G i s n o t b i p a r t i t e ;
~ [ Z E V :d(x,z)
d ( x , z ) = k, d(z,y)
= 1,
d(z,y)
= d(~,yl-kl(=
d(x,y)-llI
I
= f(d(x,y)).
)f,
0
i t can be proved by a d i r e c t argument ( c f . [ 1 6 1 ) . T h i s can a l s o be o b t a i n e d f r o m
t h e p r o o f o f Prop. 4.2, r(x,y), (zEV:
d(x,y)
-t
by r e p l a c i n g g(x,y),
+
r(x,y),
+
d(x,y)
(and by exchanging consequently t h e s e t I!(x,y)
r e s p . w i t h g(x,y), and t h e s e t
d ( x , z ) = k, d ( z , y ) = d ( x , y ) - k ) ) .
EXAMPLE 4.9.
The complete graph Kn, a t r e e G , t h e ( 2 k t l ) - c i r c u i t G a r e F-geo-
d e t i c graphs w i t h F = l . An F-geodetic graph w i t h F-1 i s c a l l e d g e o d e t i c - g r a p h ( c f . [161, [ l a ] ) . The i l k - c i r c u i t G i s F - g e o d e t i c w i t h F ( t ) = l , f o r t < k , and w i t h F ( 2 ) = 2 otherwise. EXAMPLE 4.10.
The complete b i p a r t i t e graph Kn,,=(V,E)
with V =
V'UV",
157
F-Binomial Coefficients and Related Combinatorial Topics
IV'(
=
IV"I
= n i s F-geodetic w i t h F ( 0 ) = F ( 1 ) = 1, F ( 2 ) = n.
A hypercube i s an F-graph w i t h F ( t ) = t ! . Conversely any con-
EXAMPLE 4.11.
131).
n e c t e d b i p a r t i t e F-graph w i t h F ( t ) = t ! i s a hypercube ( c f .
(cf. (51,
= t (0
The graph Kn x Km i s F - g e o d e t i c w i t h F ( t
EXAMPLE 4.12.
< t s 2 cn,m)
[151 1 -
I f Qn i s t h e n-cube, t h e graph Qn x K
EXAMPLE 4.13.
s
m
F-geodetic
with
F ( t ) = t ! ( O s t < n t l ) ( c f . [51, Prop. 3.1).
i s F - g e o d e t i c f o r some F
I f G i s a connected graph and i f GxK,
EXAMPLE 4.14.
and some m x 2 , t h e n F ( t ) = t ! . Moreover, i f G i s b i p a r t i t e , t h e n G i s a hypercube (and GxK i s a l s o a hypercube i f and o n l y i f m = 2 ) ( c f . [ 5 ] , Prop. 3.1, Cor. 3.2) m EXAMPLE 4.15. L e t G be t h e d i r e c t e d graph whose v e r t i c e s a r e t h e subspaces ~o f a g r a p h i c space o f dimension n and o f o r d e r q 3 1 and ( x , y f l a t x i s covered b y t h e f l a t y. Then
6
i s F-geodesic w i t h F ( t
i s an edge i f t h e =
[tlq!( c f .
[31,
Prop. 3.3).
REMARK 4.16.
E
L e t G be t h e u n d i r e c t e d graph a s s o c i a t e d t o t h e d i r e c t e d graph
c o n s i d e r e d i n t h e Ex. 4.15.
I n o t h e r words, G i s t h e q-analogue o f Qn-,.
Note
t h a t when q a 2, G i s n o t F-geodesic: i f x,y a r e two p o i n t s and z i s a p l a n e cont a i n i n g x, t h e n d(x,y)
= d(x,z)
t h a t t h e d i r e c t e d graph
5
graph o f Example 4.15;
=
2, b u t 2 = v(x,y)
#
u ( x , z ) = q t l . Note a l s o
o b t a i n e d f r o m G by s t a r t i n g f r o m t h e empty f l a t i s t h e
when we s t a r t from a f l a t O f 0 , t h e n
E
i s F - g e o d e t i c i f and
only i f q=l. EXAMPLE 4.17. o f Ex, 3.5.
Then
Let
6
6
=
6(P) be
t h e d i r e c t e d graph a s s o c i a t e d t o t h e p o s e t
p
i s F-geodetic a c c o r d i n g l y t o Prop, 4.1 ( b u t G i s n o t F-geode-
tic). ACKNOWLEDGEMENT.
T h i s r e s e a r c h was p a r t i a l l y supported by GNSAGA o f CNR and by
MPI.
BIBLIOGRAPHY [ 1 1 F. Buekenhout, Une c h a r a c t e r i z a t i o n des espaces a f f i n s basee s u r l a n o t i o n de d r o i t e , Math. Z. 111 (1969) 367-371. [ 21 P . V . C e c c h e r i n i , 78-98.
S u l l a nozione d i s p a z i o g r a f i c o , Rend.
Mat.
( 5 ) 6 (1967)
P.V , Ceccherini and A . Sappa
158
[ 3 ] P.V. C e c c h e r i n i , A q-analogous o f t h e c h a r a c t e r i z a t i o n o f hypercubes as graphs, J. Geometry 22 (1984) 57-74. [ 4 1 P.V. Ceccherini, A. Dragomir, Combinazioni g e n e r a l i z z a t e , q - c o e f f i c i e n t i b i n o m i a l i e spazi g r a f i c i , A t t i Convegno Geometria Combinatoria e sue a p p l i c a z i o n i (Peruaia. Settembre 1970) 137-158.
-~
[ 5 1 P.V.
Ceccherini, A. Sappa, A new c h a r a c t e r i z a t i o n o f hypercubes,Annals D i s c r e t e Math. ( t h i s volume).
[6
1 L.
[7
1 L.
C e r l i e n c o , F. P i r a s , C o e f f i c i e n t i b i n o m i a l i g e n e r a l i z z a t i , Fac. S c i . C a g l i a r i 52 (1982) 47-56.
Rend.
Sem.
C e r l i e n c o , F. Piras, G-R-Sequences and i n c i d e n c e coalgebras o f p o s e t s o f f u l l b i n o m i a l t y p e , ( t o appear).
[ 8 ] R.J. Cook, D.G. [ 9 ] H. Crapo, G.C.
Pryce, U n i f o r m l y g e o d e t i c graphs, t o appear. Rota, C o m b i n a t o r i a l geometries, MIT Press, Cambridge (1970).
[ 1 0 1 M. Deza, N.M. S i n g h i , Some p r o p e r t i e s o f p e r f e c t m a t r o i d designs, Annals D i s c r e t e Math. 6 (1980) 57-76. [ 11 ] P. D o u b i l e t , G.C. Rota, R.P. SRanley, On t h e f o u n d a t i o n s o f C o m b i n a t o r i a l t h e o r y V I : t h e i d e a o f g e n e r a t i n g f u n c t i o n , i n G.C. Rota ( e d . ) , F i n i t e Oper a t o r Calculus, Academic Press, New York (1975) 83-134. [ 12 1 J. Edmonds, U.S.R. Murti, P. Young, E q u i c a r d i n a l m a t r o i d s and m a t r o i d d e s i gns, i n "Combinatorial Mathematics and i t s A p p l i c a t i o n s " , Second Chapel H i 11 Conference ( 1 9 7 0 ) .
[ 1 3 1 S. Foldes, A 159.
c h a r a c t e r i z a t i o n o f hypercubes, D i s c r e t e Math. 17 (1977) 155-
[ 14 1 B.L. R o t h s c h i l d , N.M. S i n g h i , C h a r a c t e r i z i n g k - f l a t s i n geometric designs, J. Comb. Theory A 20 (19761, 398-403. [ 15 1 A. Sappa, C a r a t t e r i z z a z i o n e d i g r a f i t r a m i t e geodetiche, Tesi, Univ. d i Roma, D i p a r t i m e n t o d i Matematica (1984). [ 16
I R.
S c a p e l l a t o , On g e o d e t i c graphs o f diameter two and some r e l a t e d s t r u c t u r e s ( t o appear).
[ 1 7 1 B. Segre, L e c t u r e s on modern geometry. With an Appendix by L. Lombardo-Rad i c e , Cremonese, Roma (1961).
[ 1 8 ] J.C. Stempe, 266-280.
Geodetic graphs o f diameter two, J. Comb. Theory
B 17 (1974)