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General upper bound for the number of blocks having a given number of treatments common with a given block
urns1 of Statistical Piannin North-Holland Publishing
nd Inference 1 (1977) 217-234.
GENERAL UPPER BOUND FCXi THE NU.MBER HAVING A GIVEN NUMBER OF’ T TMENTS COMMON WITHAG Sanpei KAGEYAMA nnd Takumi TSUJI Received 18 October 1976; revised manuscript received 3 May 1977 Recommended by T. Calinslei T’le purpose of this paper is systematically to derive the general upper bounltl for the number of blocks having a given number of treatments common with a @en block of certain incomplete block designs. The approach adopted here is &s&on the spectral decomposition of N’N for the incidence matrix N ofa de&n, where N’ is the transpose of the matrix N. This approach will lead us to upper bounds for incomplete block designs, in particular for a large number of partially balanced incomplete block (PBIB) designs, which are not covered with the standard approach (Shah (1964 1966), Kapadia (1966)) of using well known relations Mween blocks of the designs and their association scfruemes. Several results concerning block structure of block designs are also derived from the main theorem. Finally, further 8eneralizations of the main theorem are discussed with some illustrations. AMS 1970 Subject Classifications: Primary 62K lo:, Secondary OSBOS, 62K99
Key Words and Phrases: Connected, PBIB design, PIB design, Spectra1 decomposition, AFsociatic>nscheme, ac-resolvability, Af!ine a-resolvability
P. Introduction
Shah [12] gave the upper bounds for the number of disjoint blocks in semi-regular group divisible IPBIBdesigns, certain PBIB &signs with two associate classes having a triangular association scheme, certain PBIB designs with two associate classes having an Lp association scheme and certain PBIB designs with three associate classes having a rectangular association scheme, The similar upper bound for a singular group divisible PBIB design was derived p>yKapadib LS]. Furthermore, as a generalization of the above result% Shah [M] gave the upper ‘t>ounds for the number of blocks having a given number of treatments common with a given block of the abo-de-mentionecl designs except for a singular group divisible 1.) design. All of these bounds are given by use of well-known relations between ‘blocks of designs and their association ition of N’N for the incidence matrix N of an incomplet 21’7
1u!l
/ ‘*
3
~
“^
given ‘blockof the design with ~ramete~~ 12% Iz,r rend k. ihis upper bound m6y bc: applicable to a wide &ass of block designt~.The investigtited lelasscafblock designs covers a large number of FBI3 designs, such as linked black type PIHI designs, singular and semi-regular grou,r divisible designs, certain L1designs, a particular class of Ta designs, certain PBIB , dc;sigu of the w%angulw type, all. equireplicated equibfock-sizedafine a-reso1uabi.edes@gns, srad.bhed@s cf@,&w ~~~~~~~,,?~~~~u WL Through the approach her% it will be posaibhtto +&ii bqunds ~~rni~~~t~~~~~~~~ $!Sh& and Kapadia but for FBllBdesi$ns vAich ak not cnverd with their appr@& The general upper bound derived here yields several’~omb~~atoria~~y interesting &s&s concerning block structure of block designs,Some attempt is also made to cover ewn a wider class of blook designs. The gekjeralizations are discussedwith somt illustrations, Since ZIblock design uniquely determines its incidence matrix and vicleversa, both a design and its incidencematrix are denoted by the same symbol thl aughout this paper. Further, it is supposed that all designs considered in this paper are binary, equireplicated,equiblock-sizedand connected. For the description of incompleteblack designs,RIBdesignsand PBIB designs,as well as their association schemes,we refer tn Rnghavarao [tO]. 2. General upperbound
We consider an incomplete Mock design N with parameters u, b, F and k and such that its NW has oniy one non-zero latent root, pI say, other than rk. As a spectral decomposition of YN, we can write N’N = rk
1
( >
(2.1)
g-G, +pP,
.
where ( ~/I>)c&, and P are projections corresponding to latent roots r/~ and p of NW, respectively, and G, is the h x h matrix all elenlents of which are unity. After numbering
blocks of the design as B 1, B2,. . ., .I$,, let *xicllenote the number of treatments common between B, and Bi (i = 2,3, . . ., b). Necessarily,xj’s are integers. Now, assume that d is the number of blocks having a given number I (0 $ Cg k) of treatments common with a given block of the design N. Then the first row of NW can be written ,is 45 1,I,, . ., I, = 1. Further lee the !lrst row of P be pl l, p12,. . .) -Q+2,* ,-Y4 f3 where x~==...=+.~ l
ihd+
l
I* i&d+.:**-*&be
Comparing the first rev;; of Mh sides of (2.1) we obtain p1
1 =
Mb - d/‘/h,
pls=(bC--rk),‘pb,
i==22,3,...,&
1.
(2.2)
Since (l/h )Gband P are mutually orthogonal idempotent matrices, we also obtain (2.3)
and
As A 20 from (2.4),we;have ~dl<(b-1)O-.b2Ck(r-1)-(h-l)~1. If d < b - 1 and k(r dsb-
1) # (b -
(2.5)
1)l, impIyi!rg8 > 0, then (2.5)can easily yield
b[k(rus.--__--__ l)*-/(b-l)]” _ ---k[p(b - r) + k(r2 - b)] +%if(b- l)l-2k(rl)] ’
(2.6)
Furthermore, from (2.4)it is obvious that the equality sign in (2.6)holds if and only if Pld+2=.*.= PQ,=p. In this case, from (2.1),wehave xi = rklb + p&
j=d+2,...,b.
(2.7 1
Now, since p-_ -~k(b-r)+B(bl-rk)]/[pbCb-d-
l)]
a!Id
-i- bl[(b-
l)l-2k(1~--
I)]),
Jj-l)l-k(r-
: ,){(r -- 1)bkl - y-p@- F)+ k(3 -b)]} lJ[(b - 1)1- k&- 1)]2
Hence, we can establish the follow:ng main theorem of this paper. Thearem 1. Consider an incomplete block design IV smithparameters v, b, r and k and such that its NW has only one man-zero latent root, p, other than rk. rfa giver: block of the design has d ( < 8 - 1) Bloclcshaving a given number 1 (0s 16 k) of treatments common with it, then . ufg+l__---
b[k(r-1)-l(b-1)]2 k[p(&- I’)+ k(r2 - 6)] + bl[(b - l)l- 2k-(r - 1)] ’ ------__
. cw
provided that k(r - I ) # (b - 1)l. Furthermore, ifthe equality sign in (2.8) holds, then
(k[p(b-r)+k(r”
- b)] - lbk(r - 1)}/b[k(r - 1) - l(b - 1)]
is an integer and the given block has (k[p(b-a)+k(r2
-b)]
-lbk(r--
l)}/b[k(P- l)-l(b-
1)3
treatments common wit!‘ each of the remaining (b -d - 1) blocks.
Remark 1. We now consider the case where 8 4. In this case A = - ri’[k(r - 1)Cb- 1 )a2 which is not positive. However, as A 20, we have that A 4, i.e.!, krr-l>=(b-1)l.
(2.9)
1iemx, from the definition of 0, o=bk[p(b-rj+k(r2-b)]-b2kl(r-l),
wtkh implies, from (2.9), p= k(b-r)f(b-
1).
(2.10)
221
. where t =
rank P = trace P. Relation (2.11) implies t =
b-
(2,ll)
1, which,from (2.lj, shows
that (2.12) Therefore$from (2.1),(2.10)and (2.12),we ts&ain
which implies d = b - 1 and I= k jr - 1)/(b - 1), the former indicating that N is a design of the linkdd bLck type. Thus, from a point of view of block intersection numbers, it is reasonable to assume that d < b - 1. We therefore conside&only the case of de b - 1 throughout the paper. Note that when de b - 1,620, and that when kjr- 1)# (b-1)1,8$0.
Remark 2. If k(r - 1) = (b - 1)1,it is obvious that the equality sign in (2.6)holds if and only ifd=b-1. Remark 3. The theory here is applicable to all block desigxx N that are binarty, equireplicated,equiVock-sized and coanectedl,and have only one non-zero latent rojt of NN” other than rk. But Theorem 1 is particular1.yvalid for a PBIB design N with parameters v, b, r and k and such that its NN’ has only one non-zero latent root other than rk. As applicetions of Theorem 1,we first can gentsome of the wellknown results as follows. (1) For a semi-regular group divisible PBIB design with parameters v= mn (m groupsof n treatments each), b, r, k, A,, AZ,in which rk - v1, =O, we have p = r - I1 = r(v - k)/(v- m). Applying Theorem 1we obtain Theorem 2.2 of Shah [ 145 (2) For a triangular PBIB design with parameters v=n(n- 1)/2, b, r, k, I,, iz,, and rk-vl, =It+L,)/2, in which ri-(a-41)a1 -- (n-3&=0, we have p=r-2& +A, = r(o L k)/(v - n), Applying Theorem 1 WCobtain Theorem 3.2 of Shah [ 141. (3) For an L2 PBIEBdesign with par&meters tl=s2, b, T; k, Al, A,, and rk -viz, = +-a, b in which
we have P .= f -
2a,+a~=r(v-k)/(s-1)2.
Applying Theorem 1 we obtain Theorem 4.2 of Shah [14$
(4) For a rectangular PBIB design with parameters v =ua v2 (q rows and u2 columns), E, r, k, &, I?, &,
and
we
have
Applying Theorem 1 we obtain Theorem 5.2 of Shah [14]. Furthermore, it is remarkable that Theorem 1 yields some new results which cannot be derived by Shah’s method. For example, we can obtain similar stqtements, which are routine applications of Theorem I, for the following cases. (5) A singular group divisible PBIB design with parameters v = mn and r - 1i = 0. In thiscase p-d-v& =r(v-kk)/(m1). (6) A triangular PBIB design with parameters v = n(n - 1)/2 and r - 2rll + AZ=O. in this case p=r+
(i
[n-4),1, -(il-3))L:=r(v-k)/(?2-
l)*
(7; An L, PBIB design (ii&Z) with parameters u =s2, 6, r, k, A1, AZ, and F- ii., + - 1 )i., = 0. In this case p=r+(s-i)A,
-(s-i+f1i2=r(v-k))/i(s-I).
An Li PBIB design (iz 3, with 11=s2 and r+(s-i)l,
-(s-i*f)&=O.
In this case p=r-ii,
-t-(i- l)&
==r(v-k),/(s- l)(s-i+
1).
(8) A rectangular PBIB design with r--3,, + (q -- l)(& -&)=Oand -_ -0. In this case p=r-A,f(v2-
l)(L, -12,)=)‘(u--k)/(v~ -1).
A rectangular PBIB design with
r--E,, -AZ +&
Uppep bound ofblock
intersection numbers
223
In this case
(9) A cubic PBIB design with paratnetc rs v=s3, b, r, k, Ai, AZ9&, r t (2s - 3)& =t(s-l)(s-3)Lz-(s-1)21j=QandrtIs-3)..1, -(2s-3)&+(s-l)d,=O.Inthiscase p = r(v- k)/(s - 1)3.A cubic PBIB design with-v= s3. t + (2s - 3)& + (s - 1)(s - 3)k, (s- l)*Ja=Oand r-3& +3R2 -&==O.In thiscasep=r(vk)/3(s- l)2.AcubicPBIB designwithv=s3,r+(~-~‘)11-(2s-3)~2+(s-l)&=Oandr-3il1 +3A2”&=0.1n this case p = r(v+- k)/3 (s - 1). (10) An &ne a-resolvable incomplete block design with parameters v, b ==fit, r = art and k (cf.163).In this case p = r(v - k)/(b -t ). The approach adopted here, using the spectral decomposition of NW, appears to be
usefulm a general treatment coacernifig the block structure of a design. Some of such treatments are discussed in the next section.
The following theorems and a corollary can easily be deduced from Theorem
1.
Theorem 2. If in an incomplete block desig;i the only one non-zero latent rest of NW other than rk is less thorn2r(v - k)/(b-2),
set. Proof. When
1=
then no two blocks of the design are the same
k in Theorem 1, (2.8) leads, after some calculation, to
_bp+rk-bk-p = - (b-a)k+p
’
If [bp+rkbk-p]/[(b-r)k+p] < l? which is equiv:alent to p<2rfv-k)j(b-21, then d = 0. This completes the proof. Similarly, when E= 0, from the condition that the right-hand side of (2.8) is less than 1, we obtain: Theorem 3. If in an incomplete block design the only one n’o;z-zero latertt root qf NN’ other than rk is less than k(2r2 -t b2 _Lb - 2br)/(b - r)(b - 2), then no two blocks of the design are disjoint.
Examples of Theorems 2 and 3 can easily be given for PBIB designs. Heace they are omitted here. From Theorem 1 for I = 0, we immediately obtain : Tbesrem 4. Consider an incomplete block design N with parameters v, b, r and k such that its NJ?’ has only one non-zero latent root, p, other than rk. A giver1 block cf tlw design cannot have more than b-1-----_-----_-_.
bk(r - 1)2
p(b-r)+
\(rZ-b)
[p(b - r) -t k(r2-- b)]/b(r - 1) tisan integer and aach non-di joint block has Cp(b - r)+ k(r2 - b)]Jb(r - 1) treutments common with tk given block.
This generaltheoremobviously yieldsthe followingknownresults:Theorem3.2 of Kapadia[S] fora singulargroup divi&k PBIBdesign;Theorem 2.1 of Shah[I123fora semi-regulargroupdivisible PBZPdesign;Theorem 3.1 of Shah[12] for a triangular PBIB design with rk -a 01.~=n(~&)/2; Theorem 4.1 of Shah[12] for an L2 PJHB designwith rk - vill = S(T- II ); and Theorem5.1 of Shah[12] for a rectangdarPBIIB designwith r - A1+ (vl - l)(jE2,-j23)=O~dr-dz+(oa-S)(~~-d3)t0,0fc?urse,as mentioned in Section 2, some new results different from the theorems due to Kapadia [8] and Shah [12] can easily be obtained as routine applications of Theorem 4. However, those results are omitted here. Let d* be the number of blocks disjoint with a given block. Then Theorem 4 implies d*S b- 1 -[bk(r-
1)2]/[p(b-t)+k(r2-b)].
In this case we obtain: Theorem 5. Consider an incompfcjte block design N with parameters v, b, r and k und such that its MN’ has only one non+ero latent root, p, ojmuftipficity a, other than rk. A necessary and suflcient condition for a given block @he design to have the same number of treatments common with each @‘the remaining b!ocks is that b = a + 1 and k(r - i I/ (6 - d) is an integer. In this case k(r - l)/@ - 1) is that number.
Proof. When d* = 0, i.e., f = 0 and n = 0 in Theorem 1, from (2.4) some calculation yields
which implies that b = Q+ 1 is equivalent to Pld+ 2 =. . . =plb =jk But if b is such, then P=k(b-r)/(b-l)andp=(-1)/b. Hence,from(2.7),+==k(r-l)/(b-1)+2,...,b. ts complete the proof. Examples of Theorem 5 are seen in Kageyama and Tsuji [7]. Note that p = k (b - r )/( 6 - 1) in Theorem 5 which is also a characteriza &ionof a linked block design. A dound for the number of treatments common to any two blocks can also be given by use of Tlheorem 1 as follows. Theorem 6. Consider a?a incomplete block design N with parameters v, 6, r and k and such that its NN’ has onfy one non-zero latent root, p, other than rk. The number, X, say, oftreatments common to any two blocks @the design satisfies the inequalities
Upperboundof block Intarsectlannumbws w%ere
225
I
r,=(bk(r-l)+(-1)‘+‘~br(u-k)(b-2)[(b-l)p-(b-r)k]}/b(b-l) for i=A2. ’
Pmof, ”The conditionthrr.f, the right-hand side of (2.8)in Theorem 1 is Pessthaii 1,can be
&w&en as
and so l>l,
or M,,
where Ii and I2are described in (3.1).That is, for a non-negative integer I satisfying I > II or 2< 12,there does not exist a block having a given number I of treatments common with a given block of the design. Thus the proof is completed. We now compare the bound of Theorem 6 with some bounds which are known in literature (cf. [l], [2], [3], [Sl, [13], [lS]). (i) For a BIB design with parameters v,b, r, k, and A,in whicheasep = r - A.Theorem 6 max {OJ,)$x4min
{k,l,},
(3.2)
where 1, ==[k(r-1)+,/G
k)(b-2)(r-A)@-
k)]/(b-
1)
and
Qn the other hand, it is known (cf. IS]) that the followingrelation holds; (3.3) Furthermore, Chakrabarti [S] has proved that bound (3.3) is sharper than bound (3.2). Thus, from the discussion in [S], we believe bound (3.3) to be the best for the general BIB designs. (ii) For a semi-regu lar group divisible PPiB design with parameters v = mn (m groups of n treatments each), b, r, k, A,, AZ,in which rk - v& =O, it is known (cf. [l], [2]) that the follownig relation holds :
max{0,2k-v,k+A1-r}I;xsmin
_k+r
-
a
1 *
(3.4)
S.Kcrp~?lu,
226
Tt 1suji. ’
*. )
‘,
Inthiscasep=r-A1 = r(t’-- k )/(v - nz),wit’hmultiplicity ta- M f = a. say). We shall &$w compare the upper bounds of (3.1) and (3.4). Since up = k(b - r), we have, aftersame calculation, 4-
2(rk -4 p-w--$--r
=kfb-rj{
1
,,6&$~<=ij,
(b-2)
x (b-a--l)j/ab(bA)
and ---
._-_I
l,-k=k(b-r){,/ah(b-2)(b-a-I\-abf/ab(b-1)
which imply, respectively, that
I 4 - p-k+---b-
2(rk -p)
20
ifand only if a&-
1
I
and I,--kzO
ifandonlyif
al;+&-1,
These relations show that the upper bound of (3.4) is superior trj that of (3.1). Similarly, we can show a superiority of the lower bound of (3.4). Thus, bound (3.4) is generally superior to bound (3.1). We also can compare bound (3.1) with other bounds kriown for various PBIB designs (cf. [ 11,[2]. [3], [ 131, Cl51). H owever, they are not treated here. Furthermore, bound (3.1) derived here appears to be not the best for the general BIB designs and those PBI B designs which validate Theorem 6. in addition to the above discussion, if for an incomplete block design A?all latent roots of NN’ other thal rk are equal, the design obt rously reduces to a BIB design with parameters o, b, r, k and A. Since in this case p = Y-A, Theorem 1 clearly implies: e Theorem 7. lfc1 given block ofa non-symmetric BIB design with parumeters v, b, r, k and i. has d blocks having Q given number I (OS 15 k) oJ‘treatments common with it, then
provided that ,lfr - 1) f (6 - 1)C.Furthermore, ifthe equulity sign holds, then [(r-A)k+
(;i-- l)k2 -Ik(r-
l)]/[k(r-l)-I(b-
l)]
is ~11integer and the given block has [(r-;i)k+
(i.-- 1)k2 - Ik(r - 1,]/[ k(r -- 1) -- 6(& - 1 )]
Furthermore, we get: .
Theorem 8. Consider an a-resolvable incomplete block design N with parameters V, b =fit k fl.z 2), r m&l k and such that its NW has only one non-zero latent root, of mubip!icity a, other than rk, lf’ a given block oj’ the design has /3- 1. blocks havin’g (a - I )k,‘(/$- 1) treatments common with it, then a necessarv. a~! su@ient condition&r it to have the same (though possibQ diflerent from the previous) number of’treatmenrs common with each of the remaining [b -/?I) blocks is that b = t + a and k2/v is an integer. In this case k2/v is that number.
pr(&. When for an a-resolvable incomplete block design d = /3- 1 and I= (a - 1)ic/ (fl- l), then, from (2.4) and on account of b=pt, r =:at and up = k(b- r), some calculation yields
vclhichimplies that 6= t -t-a is equivalent to pld.+ 2 =, , . =,plb =p But if b is such, then pp=Q. Hence, from (2.7), Xj = r/z/b = k2/v, j = fl + 1,. . ., 6. These results complete the proof. Examples of Theorem 8 are seen in Kageyama [63. This theorem leads to a necessary and sufficient condition for an a-resolvablle BB’I[Bdesign of the above type to be affine c+ resolvable (cf. [@I]).Furthermore, Theorem 13shows tha.t for an a-resolvable YBIB design of the above type, b 2 t + a For an incomplete block design which may be resolvablie, we have: Ttrearem 8. Consider an incomplete block design N with parumeters v = nk, b, r and k arid such that its NN’ has only one lvionctero latebzt root, oJmltltiplicity a, ot’hczrthun rk. !f’ ct given block of the design has (n - 1) blocks disjoint with it, thlen CInecessury alnd su$j%ient conditionfor it to have the same numbelr of treatments common with exch of the non-disjoint blocks is that b = r + a actd k2/v is an integer. In this case k2/v is thccl number.
S. Kugcyama, T. %ttji
225
Proof. Similarly to Theorem 5, since d* = n - 1 and ZI=nk, we have, after ele.mentary calculation, 0s
(pa;-p)‘=
i
a@-l)(r-
i=d+2
l)(b-r-a) (b-@(b-n)
-’
which implies that b = r + a :ise@Jalent to pld+ 2 =. . . =pIIQ = fi. But i1 b is such, then p = 0. Hence, from (2.7), Xj= rkJb = k2/vJ = az+ I,. ., b. These results compete the proof, This theorem leads to a necessary and sufficient condition for a resohzibk PBIB design of the above type to be rlffine resolvable (cf. [6-J).From (3S) we also have l
Corollary I& Consider an incomplete block design IV with parameters v = nk, b, r and k and such that its NM’ has only one non-zero latent root, of multiplicity a, other than rk. If s given block of the design has (ln- 1) blocks disjoint with it, thlen b 2 r + a. This corollary shows that for a resolvable PBiB design of the above ty.~ b 2 r +-a.
Examples of Theorem 9 and Corollary 10 can easily be given for affine resolvable PBIB designs and resolvable PBIB designs, respectively. In conclusion, note th+atthe main result (Theorem 1) includes many known results and novel results concerning block structure of incomplete block designs which are not necessarily PBIB designs. 4. Furthergeneralization We now consider an incomplete block design N with parameters v, b, r and k and such tLat its NN’ has on’ly two (distinct nonzero latent roots, pl, p2, say, other than rk. Note that this situation also clorresponds to a 2-associate PBIB ciesign N ofregular type, in which all the latent roots of NV’ are positive. As a spectral decomposition ofN’N we can write N’N=rk
(4.1)
where (l/b&,, Q1 and Qz are projections corresponding to latent roots rk, p1 and p2 of NW, respectively. The meaning of symbols used in this section is as follows: d is the number of blocks having a given number I (01; Is k) of treatments common with a given block. of the design. X, is the number of treatments common between the first block and the i- th block (i = 2,3,. . ., b) after numbering blocks. (k, i(,. . ., l,xd+ 2,. . ., xb) is the first row of N’N with x2 =. . . =xd+ 1 =: 1. (sl 1, s12 , . .,sJ is the first row of NW - (rk/b )G,. q n and q2 are the (1,l )-elements of matrices Ql, and Q2, respectively. Then, from (4. Pj and an argument similar to Section 2, we can get l
St 1= b
k(b - r)
(=Pr41
b b
+p2(12),
+=I=--,
rk b
i-22,3,., .,d+;,
(4.2 1
Upper boumdofblock intersection numberg
Writing S= (g =,$+2s:. )/‘(a
229
- (a!- 1 ), from ~(4.2) we obtain
2
. (4.31
--
and so
whichimplies
where
e=b(p:ql +p;q2)+k2(r2-b)+bZ[(b-
1&2k(r-
l)].
If d < b - 1 and k(r - 1) f (b - 1 )l, implying 0 r 0, then (4.4) yields i!jb-1-b[k(r-1)-(b-1)~2/8.
(4.5 )
Note that 6 depends on values of q1 and q2 which vary according to zhe choice of a given block of the design. More generally, timre are relations, Qi = (l/pi)N’PiN, where P,‘s are projections corresponding to nonzero latent roots of NW for i = 1,2. Now, when pi >p2, from (4.2) we get
which implies 8sk[p,(b--r)+k(r2
-b)]-t-bl[(b-l)Z-2k(r-I)].
Hence?from (4.3, we r;an obtain ,
dgb-I--
b[k(r -- f ) - (b a- 1 )Z”j2 PWll__ k[pl (!J- I‘)+ k(r2 - h)] -I-.
&b-l
h[k(r-!i-(b-l)q2
------
(4.7)
k[p2(b- r’)+ k(r2-b)].tL!C4b-1)~-ak(~-~~’
where p2 3 p1 > 0. Furtlrermlore, from (4.3) it is obvious that the equality sign in (4.5) holds if and only if - S. In this case, from ?he defmition of xj and 81sfollowing (4.1),we have Sld+p=...=SlP,x,==r.k/b+$,
j=d+&..
$%.8)
,b.
Now, as 6= -I:k(b-r)+d(bf-rk)]/~(b-d-l),fromi4.8) s
_b(p:ql .-
1
+p;q2)+k2(F2--b)-bk-l(r--1) hEko_ (b - 1 )lJ
we alsoget ?
j=d+2,...,b
Thus, WCcan establish : Theorem 11. Consider un incomplete Hock design iV with parameters v, b, r and k and such that its NN’ has on/y two distinct wxxcro latent roots, pt, p2, other than rk. If a given block of the design has d ( < b -- 1) blocks having a given sumber 1 (0 s & k) of
treatments common with it, then P[k(r-l)-(b-1)~2
agl-l-
_LI-
HP:41
+ Pi42 I+
kZIr2 --4)+bl[(b-
l)l-2k(r-1)1’
(4.9)
provided thut k( r - 1) f (b - 1 )l. Furthermore, ifthe equality sigrvin (4.9) holds, then INP:y,
+ PSI2
‘+-k2(r2-b)-bkl(r-l)}/{b[k(r-l)-(b-1)0}
is an integer and the given block has {b(p&
+p$42)+k2(r’!
--b)-bkZ(r-
l))/{b[k(r-l)-
(b- l)g}
treatments common with each ofthe remaining (6 -d - 1) blocks. Theorem 11 is a generalization of Theorem I. In fact, when p1 =p2 ( = p9 say), from (4.2) we obtain b(pfqi epiq2) ==pk(b - r) which shows that Theorem 11 yields Theorem I when pl = p2* Note that bounds (4.6) and (4.7) are of the same form as (2.8)
in Theorem I. I-Iowever, (4.6) and (4.7) are derived for a design N with only two nonzero latent roots of NN’ other than rk. IOnthe other hand, (2.8) is derived for a design N with only one con-zero latent root ‘ofNN’ other than rk. Bound (4.9)is represented in terins of parameters of the design I’, latent roots of N N’ and values of 4;s which vary according to the choice of a given block, Because of this fact it may not be easy to evaluate (4.$). On the other hand, bounds (4.6) and (4.7) are conveniently represented in terms of parameters of the design and the maximum latent
@
where p1 ==rk-vUAq, p2=+A1, ~=I,-(l/n)tl,@G,) and ,4@B is the Kronecker product of the matries A =[a,l and B, i.e., A@B= [a$?). Let ai be the number of treatm,ents ccrnmon between a given block of the PBIB design and the i-th group of the ass&&on scheme for i = 1,2,. . ., m. Then from Qz = ( llpz ) N’PN we can obtain
which, from (4.2),yields (4.11) We now consider a regular group divisible PBIB design R19([4]) with parameters u=12,b=32,r=:8,k==3,A,=2,d2=1 andwith32blocks,(1,3,7)(3,5,9)(5,7,11)(1,7, 9) (3,9,11) (1,$11)(2,4,8) (4,6,10) (6,8,12) (2,8,lVj (4,10,12) (2,6,12) (1,3,4) (2,7,9) (3,5,12)(6,9,101(:1,6,8)(2,4,11)(3,7,8)(4,9,12)(5,7,10)(1,10,11)(2,5,6)(3,6, 11)(8, 9,ll) (4,5,8) (4,6,7) (7, i 1,12) (8,10,12) (1,2,12) (2,3,10) (1,5,9), which iniply p1 .-=12 and p2 =6 (pl > p& If the first block (1, 3,7) is chosen as a given block, then y1 = l/l6 and q2 = 1/4from i14.10)and (4.11). For the 13.th block. (1,3,4) we get q, := l/144 a.lldy2 = l3/36. In these cases, the values (by Gauss symbcl) of tl:e right-hand Gde of bounds (4.6) and (4.9) and1the exact values 3.’d in the design are tabulated for etach !(O< I I k = 3) in Tables 1 ad 2, Table 2 appars to show that bound (4,9) satisfactorily rifliits the existing block structure of the design concerning the chosen block (1,3,4). We can produce tables similar to the above for other regular group divisible PBIB designs by * use of (4.10) and @Al). However, they are omitt8d here. Example (ii) Clonsider a 2-ass&ate ‘PBIB design N with parameters u ,-cn(n - 1)/2, b, r, k, Al, A2baseci on a triangular association scheme with association ‘matrices AO,A I and &. Thm it is known (cf. [7]) that the spectral decompocitior of NW is
wh8re
232
--i
0
t
2
3
27 24 1s --.
8 5 3
3 2 0
11
2
3
8 *4 1
3 1 0
;
a.----
(4.6) (4.9) d --
18 14 13
.
Tabie 2 Chosen block (1,3,4) 0
1
.(4.4) (4.9 1 d
18 11 11
27 22 19 -.
and
Let ui be the number of treatments common between a given block of the PBIB design and the i-th row of the association scheme for i = I, 2,. . ., n. We then obtain, from Q1 = (l/pl)N’,PIN, (4.12) which, from (4.2)?yields (4.13) We now consider a triangular PBIB design Tl4([4]) with parameters v = 14 h = 20, r =6,k=3,~,=1,A2=2andvvith20blocks,(1,2,10)(1,5,10)(2~5,lO)(1,8,9)(2,7,8)(2,. 337)(3,7,8~ (1,3,4) f2,6,9’) (2,4,6) (4,6,9) (4, ! 8) (4,810) (1,7,9) (1,6,8) (3,599) (399, 10)(4,5,7) (3,5,6) (6,7, lo), whiichimply pr - 3 and p2 = 6 (p2 r pl). If the first block (1, 2, 10) is chosen as a given block, then q1 -4,45 and q2 - 11,/36from (4.12) andj4.13). For the last block (6,7, lo) we get q1 :r=8/l 5 and q2 = l/12. In these cases, the values (by Gauss’ symbol) of the right hand side Qfbounds (4.7) and (4.9) and the exact values of d in the design are tabulated for each I (0 s I ;l k = 3) in Tables 3 and 4, Table 4 appears to show that bound (4.!3)satisfactorily reflects the existing block structure of the design concerning the chiosen b”aock(6,7, lo), We can also produce tables similar to the above for other triangular PBIB designs of regular type by u.se of (4.12) and (4.131).However, they are not described here.
, .
Upper bsund of block intersection numbers
233
Table4 ChosenMock (6,7,10) I
0
1
2
3
WI
7 4 4
17 1s 15
4 1 0
1 ‘4 0
(49) ‘J
Furthermore, for other PBIB designs based on various association schemes we can produce these tables, but they are omitted here. Acknowledgments This research was partially supported by the Grant 163012 in Aid for General Research (D) of the Ministry of Education, Japan. The authors wish to thank the referee for his useful comments. References lfl] Agrawal, H.L. (1964). On the bounds of the number ofcommon treatments between blocks of certain two associate PBIB designs. Calcutta Statist. Assoc. Bull. 13,76-79. [23 Agrawal,H.L. (1964). Qn the bounds of the number of common treatments between blocks of semiregular group divisible designs, J. Am. Stat&. Aunsoc,59,867~87 1. 133 Agrawal, H.L. (1966). Comparison of the bounds of the number of common treatments between blocks of certainpartially balancedincompleteblock,designs,Ann. Mnth. Stcltist. 37,739- 740. [4] BOSGkc., H. Clarworthy and S.S. Shrikhande (1954). Tables of partially balanced incomplete block designs with two associate classes. North Carolina Agricultural Experimental Bulletin, No. 107. [S] ChaksGarti, MC. (1963). Query. J, Indian Statist. Assoc. 1,230-234. [6] Kageyama, S. (1977). Conditions for a-resolvabilit,y and aftme c+resolvability of incomplete block designs. 1. Japan Statist, Sot. 7, W-25. [7] Kageyama S. and T. Tsuji. Characterization of certain incomplete block designs. J.S.P.I. 1. [8] Kapadia,C.H. (1966). On the bloc% structure of singular group divisible designs. tlnn. Math. Stat%. 37, 1398- 1400, [9] Majindar, RN. (1962). On the parameters and intersection of blocks of balanced inr:omplete block des@ts. ALnn,Math. Statist. 33,12OU-1205. [lo] Raghavarao, D. (1971). Constructions and dombhatorial Problems in Des@ of Experiwzts. John Wiley, New York. ’ B
[ 113. !Shah,S.M. (196.3).On the uppr bound for the number of blocksin balanced incomplete blmk dqipass
having a given numb of treatwants common with a given block. J. Indian S~ist.&soc. 1,2H-220. Sh:nh,S.M. (1964).An upper by ula#for the number of disjoint blocks in cett& I?B!B @signs: Aqw ’ l.L Malk Statist. 35,39&407. Shah, S.M. (1963).Bounds fok the number of commo&eatments b&w&n an+ tie blocks $&r&in PBIB designs.Am. Math Statist. 36,33?-342. Shah, S.M. (1966).On the block structure,of certain #artiaii$ balanced iacomplcta bloik dosip. Ann, Marh. Statist. 3?,1016-1020. Surendlran,P.U. (1968).Cotuman treatments between blocks of certain parti@ balanced . r incomplete bllockdesigns.Ann.Math. Statist. 39.999-1006.
nd Inference 1 (1977) 217-234.
GENERAL UPPER BOUND FCXi THE NU.MBER HAVING A GIVEN NUMBER OF’ T TMENTS COMMON WITHAG Sanpei KAGEYAMA nnd Takumi TSUJI Received 18 October 1976; revised manuscript received 3 May 1977 Recommended by T. Calinslei T’le purpose of this paper is systematically to derive the general upper bounltl for the number of blocks having a given number of treatments common with a @en block of certain incomplete block designs. The approach adopted here is &s&on the spectral decomposition of N’N for the incidence matrix N ofa de&n, where N’ is the transpose of the matrix N. This approach will lead us to upper bounds for incomplete block designs, in particular for a large number of partially balanced incomplete block (PBIB) designs, which are not covered with the standard approach (Shah (1964 1966), Kapadia (1966)) of using well known relations Mween blocks of the designs and their association scfruemes. Several results concerning block structure of block designs are also derived from the main theorem. Finally, further 8eneralizations of the main theorem are discussed with some illustrations. AMS 1970 Subject Classifications: Primary 62K lo:, Secondary OSBOS, 62K99
Key Words and Phrases: Connected, PBIB design, PIB design, Spectra1 decomposition, AFsociatic>nscheme, ac-resolvability, Af!ine a-resolvability
P. Introduction
Shah [12] gave the upper bounds for the number of disjoint blocks in semi-regular group divisible IPBIBdesigns, certain PBIB &signs with two associate classes having a triangular association scheme, certain PBIB designs with two associate classes having an Lp association scheme and certain PBIB designs with three associate classes having a rectangular association scheme, The similar upper bound for a singular group divisible PBIB design was derived p>yKapadib LS]. Furthermore, as a generalization of the above result% Shah [M] gave the upper ‘t>ounds for the number of blocks having a given number of treatments common with a given block of the abo-de-mentionecl designs except for a singular group divisible 1.) design. All of these bounds are given by use of well-known relations between ‘blocks of designs and their association ition of N’N for the incidence matrix N of an incomplet 21’7
1u!l
/ ‘*
3
~
“^
given ‘blockof the design with ~ramete~~ 12% Iz,r rend k. ihis upper bound m6y bc: applicable to a wide &ass of block designt~.The investigtited lelasscafblock designs covers a large number of FBI3 designs, such as linked black type PIHI designs, singular and semi-regular grou,r divisible designs, certain L1designs, a particular class of Ta designs, certain PBIB , dc;sigu of the w%angulw type, all. equireplicated equibfock-sizedafine a-reso1uabi.edes@gns, srad.bhed@s cf@,&w ~~~~~~~,,?~~~~u WL Through the approach her% it will be posaibhtto +&ii bqunds ~~rni~~~t~~~~~~~~ $!Sh& and Kapadia but for FBllBdesi$ns vAich ak not cnverd with their appr@& The general upper bound derived here yields several’~omb~~atoria~~y interesting &s&s concerning block structure of block designs,Some attempt is also made to cover ewn a wider class of blook designs. The gekjeralizations are discussedwith somt illustrations, Since ZIblock design uniquely determines its incidence matrix and vicleversa, both a design and its incidencematrix are denoted by the same symbol thl aughout this paper. Further, it is supposed that all designs considered in this paper are binary, equireplicated,equiblock-sizedand connected. For the description of incompleteblack designs,RIBdesignsand PBIB designs,as well as their association schemes,we refer tn Rnghavarao [tO]. 2. General upperbound
We consider an incomplete Mock design N with parameters u, b, F and k and such that its NW has oniy one non-zero latent root, pI say, other than rk. As a spectral decomposition of YN, we can write N’N = rk
1
( >
(2.1)
g-G, +pP,
.
where ( ~/I>)c&, and P are projections corresponding to latent roots r/~ and p of NW, respectively, and G, is the h x h matrix all elenlents of which are unity. After numbering
blocks of the design as B 1, B2,. . ., .I$,, let *xicllenote the number of treatments common between B, and Bi (i = 2,3, . . ., b). Necessarily,xj’s are integers. Now, assume that d is the number of blocks having a given number I (0 $ Cg k) of treatments common with a given block of the design N. Then the first row of NW can be written ,is 45 1,I,, . ., I, = 1. Further lee the !lrst row of P be pl l, p12,. . .) -Q+2,* ,-Y4 f3 where x~==...=+.~ l
ihd+
l
I* i&d+.:**-*&be
Comparing the first rev;; of Mh sides of (2.1) we obtain p1
1 =
Mb - d/‘/h,
pls=(bC--rk),‘pb,
i==22,3,...,&
1.
(2.2)
Since (l/h )Gband P are mutually orthogonal idempotent matrices, we also obtain (2.3)
and
As A 20 from (2.4),we;have ~dl<(b-1)O-.b2Ck(r-1)-(h-l)~1. If d < b - 1 and k(r dsb-
1) # (b -
(2.5)
1)l, impIyi!rg8 > 0, then (2.5)can easily yield
b[k(rus.--__--__ l)*-/(b-l)]” _ ---k[p(b - r) + k(r2 - b)] +%if(b- l)l-2k(rl)] ’
(2.6)
Furthermore, from (2.4)it is obvious that the equality sign in (2.6)holds if and only if Pld+2=.*.= PQ,=p. In this case, from (2.1),wehave xi = rklb + p&
j=d+2,...,b.
(2.7 1
Now, since p-_ -~k(b-r)+B(bl-rk)]/[pbCb-d-
l)]
a!Id
-i- bl[(b-
l)l-2k(1~--
I)]),
Jj-l)l-k(r-
: ,){(r -- 1)bkl - y-p@- F)+ k(3 -b)]} lJ[(b - 1)1- k&- 1)]2
Hence, we can establish the follow:ng main theorem of this paper. Thearem 1. Consider an incomplete block design IV smithparameters v, b, r and k and such that its NW has only one man-zero latent root, p, other than rk. rfa giver: block of the design has d ( < 8 - 1) Bloclcshaving a given number 1 (0s 16 k) of treatments common with it, then . ufg+l__---
b[k(r-1)-l(b-1)]2 k[p(&- I’)+ k(r2 - 6)] + bl[(b - l)l- 2k-(r - 1)] ’ ------__
. cw
provided that k(r - I ) # (b - 1)l. Furthermore, ifthe equality sign in (2.8) holds, then
(k[p(b-r)+k(r”
- b)] - lbk(r - 1)}/b[k(r - 1) - l(b - 1)]
is an integer and the given block has (k[p(b-a)+k(r2
-b)]
-lbk(r--
l)}/b[k(P- l)-l(b-
1)3
treatments common wit!‘ each of the remaining (b -d - 1) blocks.
Remark 1. We now consider the case where 8 4. In this case A = - ri’[k(r - 1)Cb- 1 )a2 which is not positive. However, as A 20, we have that A 4, i.e.!, krr-l>=(b-1)l.
(2.9)
1iemx, from the definition of 0, o=bk[p(b-rj+k(r2-b)]-b2kl(r-l),
wtkh implies, from (2.9), p= k(b-r)f(b-
1).
(2.10)
221
. where t =
rank P = trace P. Relation (2.11) implies t =
b-
(2,ll)
1, which,from (2.lj, shows
that (2.12) Therefore$from (2.1),(2.10)and (2.12),we ts&ain
which implies d = b - 1 and I= k jr - 1)/(b - 1), the former indicating that N is a design of the linkdd bLck type. Thus, from a point of view of block intersection numbers, it is reasonable to assume that d < b - 1. We therefore conside&only the case of de b - 1 throughout the paper. Note that when de b - 1,620, and that when kjr- 1)# (b-1)1,8$0.
Remark 2. If k(r - 1) = (b - 1)1,it is obvious that the equality sign in (2.6)holds if and only ifd=b-1. Remark 3. The theory here is applicable to all block desigxx N that are binarty, equireplicated,equiVock-sized and coanectedl,and have only one non-zero latent rojt of NN” other than rk. But Theorem 1 is particular1.yvalid for a PBIB design N with parameters v, b, r and k and such that its NN’ has only one non-zero latent root other than rk. As applicetions of Theorem 1,we first can gentsome of the wellknown results as follows. (1) For a semi-regular group divisible PBIB design with parameters v= mn (m groupsof n treatments each), b, r, k, A,, AZ,in which rk - v1, =O, we have p = r - I1 = r(v - k)/(v- m). Applying Theorem 1we obtain Theorem 2.2 of Shah [ 145 (2) For a triangular PBIB design with parameters v=n(n- 1)/2, b, r, k, I,, iz,, and rk-vl, =It+L,)/2, in which ri-(a-41)a1 -- (n-3&=0, we have p=r-2& +A, = r(o L k)/(v - n), Applying Theorem 1 WCobtain Theorem 3.2 of Shah [ 141. (3) For an L2 PBIEBdesign with par&meters tl=s2, b, T; k, Al, A,, and rk -viz, = +-a, b in which
we have P .= f -
2a,+a~=r(v-k)/(s-1)2.
Applying Theorem 1 we obtain Theorem 4.2 of Shah [14$
(4) For a rectangular PBIB design with parameters v =ua v2 (q rows and u2 columns), E, r, k, &, I?, &,
and
we
have
Applying Theorem 1 we obtain Theorem 5.2 of Shah [14]. Furthermore, it is remarkable that Theorem 1 yields some new results which cannot be derived by Shah’s method. For example, we can obtain similar stqtements, which are routine applications of Theorem I, for the following cases. (5) A singular group divisible PBIB design with parameters v = mn and r - 1i = 0. In thiscase p-d-v& =r(v-kk)/(m1). (6) A triangular PBIB design with parameters v = n(n - 1)/2 and r - 2rll + AZ=O. in this case p=r+
(i
[n-4),1, -(il-3))L:=r(v-k)/(?2-
l)*
(7; An L, PBIB design (ii&Z) with parameters u =s2, 6, r, k, A1, AZ, and F- ii., + - 1 )i., = 0. In this case p=r+(s-i)A,
-(s-i+f1i2=r(v-k))/i(s-I).
An Li PBIB design (iz 3, with 11=s2 and r+(s-i)l,
-(s-i*f)&=O.
In this case p=r-ii,
-t-(i- l)&
==r(v-k),/(s- l)(s-i+
1).
(8) A rectangular PBIB design with r--3,, + (q -- l)(& -&)=Oand -_ -0. In this case p=r-A,f(v2-
l)(L, -12,)=)‘(u--k)/(v~ -1).
A rectangular PBIB design with
r--E,, -AZ +&
Uppep bound ofblock
intersection numbers
223
In this case
(9) A cubic PBIB design with paratnetc rs v=s3, b, r, k, Ai, AZ9&, r t (2s - 3)& =t(s-l)(s-3)Lz-(s-1)21j=QandrtIs-3)..1, -(2s-3)&+(s-l)d,=O.Inthiscase p = r(v- k)/(s - 1)3.A cubic PBIB design with-v= s3. t + (2s - 3)& + (s - 1)(s - 3)k, (s- l)*Ja=Oand r-3& +3R2 -&==O.In thiscasep=r(vk)/3(s- l)2.AcubicPBIB designwithv=s3,r+(~-~‘)11-(2s-3)~2+(s-l)&=Oandr-3il1 +3A2”&=0.1n this case p = r(v+- k)/3 (s - 1). (10) An &ne a-resolvable incomplete block design with parameters v, b ==fit, r = art and k (cf.163).In this case p = r(v - k)/(b -t ). The approach adopted here, using the spectral decomposition of NW, appears to be
usefulm a general treatment coacernifig the block structure of a design. Some of such treatments are discussed in the next section.
The following theorems and a corollary can easily be deduced from Theorem
1.
Theorem 2. If in an incomplete block desig;i the only one non-zero latent rest of NW other than rk is less thorn2r(v - k)/(b-2),
set. Proof. When
1=
then no two blocks of the design are the same
k in Theorem 1, (2.8) leads, after some calculation, to
_bp+rk-bk-p = - (b-a)k+p
’
If [bp+rkbk-p]/[(b-r)k+p] < l? which is equiv:alent to p<2rfv-k)j(b-21, then d = 0. This completes the proof. Similarly, when E= 0, from the condition that the right-hand side of (2.8) is less than 1, we obtain: Theorem 3. If in an incomplete block design the only one n’o;z-zero latertt root qf NN’ other than rk is less than k(2r2 -t b2 _Lb - 2br)/(b - r)(b - 2), then no two blocks of the design are disjoint.
Examples of Theorems 2 and 3 can easily be given for PBIB designs. Heace they are omitted here. From Theorem 1 for I = 0, we immediately obtain : Tbesrem 4. Consider an incomplete block design N with parameters v, b, r and k such that its NJ?’ has only one non-zero latent root, p, other than rk. A giver1 block cf tlw design cannot have more than b-1-----_-----_-_.
bk(r - 1)2
p(b-r)+
\(rZ-b)
[p(b - r) -t k(r2-- b)]/b(r - 1) tisan integer and aach non-di joint block has Cp(b - r)+ k(r2 - b)]Jb(r - 1) treutments common with tk given block.
This generaltheoremobviously yieldsthe followingknownresults:Theorem3.2 of Kapadia[S] fora singulargroup divi&k PBIBdesign;Theorem 2.1 of Shah[I123fora semi-regulargroupdivisible PBZPdesign;Theorem 3.1 of Shah[12] for a triangular PBIB design with rk -a 01.~=n(~&)/2; Theorem 4.1 of Shah[12] for an L2 PJHB designwith rk - vill = S(T- II ); and Theorem5.1 of Shah[12] for a rectangdarPBIIB designwith r - A1+ (vl - l)(jE2,-j23)=O~dr-dz+(oa-S)(~~-d3)t0,0fc?urse,as mentioned in Section 2, some new results different from the theorems due to Kapadia [8] and Shah [12] can easily be obtained as routine applications of Theorem 4. However, those results are omitted here. Let d* be the number of blocks disjoint with a given block. Then Theorem 4 implies d*S b- 1 -[bk(r-
1)2]/[p(b-t)+k(r2-b)].
In this case we obtain: Theorem 5. Consider an incompfcjte block design N with parameters v, b, r and k und such that its MN’ has only one non+ero latent root, p, ojmuftipficity a, other than rk. A necessary and suflcient condition for a given block @he design to have the same number of treatments common with each @‘the remaining b!ocks is that b = a + 1 and k(r - i I/ (6 - d) is an integer. In this case k(r - l)/@ - 1) is that number.
Proof. When d* = 0, i.e., f = 0 and n = 0 in Theorem 1, from (2.4) some calculation yields
which implies that b = Q+ 1 is equivalent to Pld+ 2 =. . . =plb =jk But if b is such, then P=k(b-r)/(b-l)andp=(-1)/b. Hence,from(2.7),+==k(r-l)/(b-1)+2,...,b. ts complete the proof. Examples of Theorem 5 are seen in Kageyama and Tsuji [7]. Note that p = k (b - r )/( 6 - 1) in Theorem 5 which is also a characteriza &ionof a linked block design. A dound for the number of treatments common to any two blocks can also be given by use of Tlheorem 1 as follows. Theorem 6. Consider a?a incomplete block design N with parameters v, 6, r and k and such that its NN’ has onfy one non-zero latent root, p, other than rk. The number, X, say, oftreatments common to any two blocks @the design satisfies the inequalities
Upperboundof block Intarsectlannumbws w%ere
225
I
r,=(bk(r-l)+(-1)‘+‘~br(u-k)(b-2)[(b-l)p-(b-r)k]}/b(b-l) for i=A2. ’
Pmof, ”The conditionthrr.f, the right-hand side of (2.8)in Theorem 1 is Pessthaii 1,can be
&w&en as
and so l>l,
or M,,
where Ii and I2are described in (3.1).That is, for a non-negative integer I satisfying I > II or 2< 12,there does not exist a block having a given number I of treatments common with a given block of the design. Thus the proof is completed. We now compare the bound of Theorem 6 with some bounds which are known in literature (cf. [l], [2], [3], [Sl, [13], [lS]). (i) For a BIB design with parameters v,b, r, k, and A,in whicheasep = r - A.Theorem 6 max {OJ,)$x4min
{k,l,},
(3.2)
where 1, ==[k(r-1)+,/G
k)(b-2)(r-A)@-
k)]/(b-
1)
and
Qn the other hand, it is known (cf. IS]) that the followingrelation holds; (3.3) Furthermore, Chakrabarti [S] has proved that bound (3.3) is sharper than bound (3.2). Thus, from the discussion in [S], we believe bound (3.3) to be the best for the general BIB designs. (ii) For a semi-regu lar group divisible PPiB design with parameters v = mn (m groups of n treatments each), b, r, k, A,, AZ,in which rk - v& =O, it is known (cf. [l], [2]) that the follownig relation holds :
max{0,2k-v,k+A1-r}I;xsmin
_k+r
-
a
1 *
(3.4)
S.Kcrp~?lu,
226
Tt 1suji. ’
*. )
‘,
Inthiscasep=r-A1 = r(t’-- k )/(v - nz),wit’hmultiplicity ta- M f = a. say). We shall &$w compare the upper bounds of (3.1) and (3.4). Since up = k(b - r), we have, aftersame calculation, 4-
2(rk -4 p-w--$--r
=kfb-rj{
1
,,6&$~<=ij,
(b-2)
x (b-a--l)j/ab(bA)
and ---
._-_I
l,-k=k(b-r){,/ah(b-2)(b-a-I\-abf/ab(b-1)
which imply, respectively, that
I 4 - p-k+---b-
2(rk -p)
20
ifand only if a&-
1
I
and I,--kzO
ifandonlyif
al;+&-1,
These relations show that the upper bound of (3.4) is superior trj that of (3.1). Similarly, we can show a superiority of the lower bound of (3.4). Thus, bound (3.4) is generally superior to bound (3.1). We also can compare bound (3.1) with other bounds kriown for various PBIB designs (cf. [ 11,[2]. [3], [ 131, Cl51). H owever, they are not treated here. Furthermore, bound (3.1) derived here appears to be not the best for the general BIB designs and those PBI B designs which validate Theorem 6. in addition to the above discussion, if for an incomplete block design A?all latent roots of NN’ other thal rk are equal, the design obt rously reduces to a BIB design with parameters o, b, r, k and A. Since in this case p = Y-A, Theorem 1 clearly implies: e Theorem 7. lfc1 given block ofa non-symmetric BIB design with parumeters v, b, r, k and i. has d blocks having Q given number I (OS 15 k) oJ‘treatments common with it, then
provided that ,lfr - 1) f (6 - 1)C.Furthermore, ifthe equulity sign holds, then [(r-A)k+
(;i-- l)k2 -Ik(r-
l)]/[k(r-l)-I(b-
l)]
is ~11integer and the given block has [(r-;i)k+
(i.-- 1)k2 - Ik(r - 1,]/[ k(r -- 1) -- 6(& - 1 )]
Furthermore, we get: .
Theorem 8. Consider an a-resolvable incomplete block design N with parameters V, b =fit k fl.z 2), r m&l k and such that its NW has only one non-zero latent root, of mubip!icity a, other than rk, lf’ a given block oj’ the design has /3- 1. blocks havin’g (a - I )k,‘(/$- 1) treatments common with it, then a necessarv. a~! su@ient condition&r it to have the same (though possibQ diflerent from the previous) number of’treatmenrs common with each of the remaining [b -/?I) blocks is that b = t + a and k2/v is an integer. In this case k2/v is that number.
pr(&. When for an a-resolvable incomplete block design d = /3- 1 and I= (a - 1)ic/ (fl- l), then, from (2.4) and on account of b=pt, r =:at and up = k(b- r), some calculation yields
vclhichimplies that 6= t -t-a is equivalent to pld.+ 2 =, , . =,plb =p But if b is such, then pp=Q. Hence, from (2.7), Xj = r/z/b = k2/v, j = fl + 1,. . ., 6. These results complete the proof. Examples of Theorem 8 are seen in Kageyama [63. This theorem leads to a necessary and sufficient condition for an a-resolvablle BB’I[Bdesign of the above type to be affine c+ resolvable (cf. [@I]).Furthermore, Theorem 13shows tha.t for an a-resolvable YBIB design of the above type, b 2 t + a For an incomplete block design which may be resolvablie, we have: Ttrearem 8. Consider an incomplete block design N with parumeters v = nk, b, r and k arid such that its NN’ has only one lvionctero latebzt root, oJmltltiplicity a, ot’hczrthun rk. !f’ ct given block of the design has (n - 1) blocks disjoint with it, thlen CInecessury alnd su$j%ient conditionfor it to have the same numbelr of treatments common with exch of the non-disjoint blocks is that b = r + a actd k2/v is an integer. In this case k2/v is thccl number.
S. Kugcyama, T. %ttji
225
Proof. Similarly to Theorem 5, since d* = n - 1 and ZI=nk, we have, after ele.mentary calculation, 0s
(pa;-p)‘=
i
a@-l)(r-
i=d+2
l)(b-r-a) (b-@(b-n)
-’
which implies that b = r + a :ise@Jalent to pld+ 2 =. . . =pIIQ = fi. But i1 b is such, then p = 0. Hence, from (2.7), Xj= rkJb = k2/vJ = az+ I,. ., b. These results compete the proof, This theorem leads to a necessary and sufficient condition for a resohzibk PBIB design of the above type to be rlffine resolvable (cf. [6-J).From (3S) we also have l
Corollary I& Consider an incomplete block design IV with parameters v = nk, b, r and k and such that its NM’ has only one non-zero latent root, of multiplicity a, other than rk. If s given block of the design has (ln- 1) blocks disjoint with it, thlen b 2 r + a. This corollary shows that for a resolvable PBiB design of the above ty.~ b 2 r +-a.
Examples of Theorem 9 and Corollary 10 can easily be given for affine resolvable PBIB designs and resolvable PBIB designs, respectively. In conclusion, note th+atthe main result (Theorem 1) includes many known results and novel results concerning block structure of incomplete block designs which are not necessarily PBIB designs. 4. Furthergeneralization We now consider an incomplete block design N with parameters v, b, r and k and such tLat its NN’ has on’ly two (distinct nonzero latent roots, pl, p2, say, other than rk. Note that this situation also clorresponds to a 2-associate PBIB ciesign N ofregular type, in which all the latent roots of NV’ are positive. As a spectral decomposition ofN’N we can write N’N=rk
(4.1)
where (l/b&,, Q1 and Qz are projections corresponding to latent roots rk, p1 and p2 of NW, respectively. The meaning of symbols used in this section is as follows: d is the number of blocks having a given number I (01; Is k) of treatments common with a given block. of the design. X, is the number of treatments common between the first block and the i- th block (i = 2,3,. . ., b) after numbering blocks. (k, i(,. . ., l,xd+ 2,. . ., xb) is the first row of N’N with x2 =. . . =xd+ 1 =: 1. (sl 1, s12 , . .,sJ is the first row of NW - (rk/b )G,. q n and q2 are the (1,l )-elements of matrices Ql, and Q2, respectively. Then, from (4. Pj and an argument similar to Section 2, we can get l
St 1= b
k(b - r)
(=Pr41
b b
+p2(12),
+=I=--,
rk b
i-22,3,., .,d+;,
(4.2 1
Upper boumdofblock intersection numberg
Writing S= (g =,$+2s:. )/‘(a
229
- (a!- 1 ), from ~(4.2) we obtain
2
. (4.31
--
and so
whichimplies
where
e=b(p:ql +p;q2)+k2(r2-b)+bZ[(b-
1&2k(r-
l)].
If d < b - 1 and k(r - 1) f (b - 1 )l, implying 0 r 0, then (4.4) yields i!jb-1-b[k(r-1)-(b-1)~2/8.
(4.5 )
Note that 6 depends on values of q1 and q2 which vary according to zhe choice of a given block of the design. More generally, timre are relations, Qi = (l/pi)N’PiN, where P,‘s are projections corresponding to nonzero latent roots of NW for i = 1,2. Now, when pi >p2, from (4.2) we get
which implies 8sk[p,(b--r)+k(r2
-b)]-t-bl[(b-l)Z-2k(r-I)].
Hence?from (4.3, we r;an obtain ,
dgb-I--
b[k(r -- f ) - (b a- 1 )Z”j2 PWll__ k[pl (!J- I‘)+ k(r2 - h)] -I-.
&b-l
h[k(r-!i-(b-l)q2
------
(4.7)
k[p2(b- r’)+ k(r2-b)].tL!C4b-1)~-ak(~-~~’
where p2 3 p1 > 0. Furtlrermlore, from (4.3) it is obvious that the equality sign in (4.5) holds if and only if - S. In this case, from ?he defmition of xj and 81sfollowing (4.1),we have Sld+p=...=SlP,x,==r.k/b+$,
j=d+&..
$%.8)
,b.
Now, as 6= -I:k(b-r)+d(bf-rk)]/~(b-d-l),fromi4.8) s
_b(p:ql .-
1
+p;q2)+k2(F2--b)-bk-l(r--1) hEko_ (b - 1 )lJ
we alsoget ?
j=d+2,...,b
Thus, WCcan establish : Theorem 11. Consider un incomplete Hock design iV with parameters v, b, r and k and such that its NN’ has on/y two distinct wxxcro latent roots, pt, p2, other than rk. If a given block of the design has d ( < b -- 1) blocks having a given sumber 1 (0 s & k) of
treatments common with it, then P[k(r-l)-(b-1)~2
agl-l-
_LI-
HP:41
+ Pi42 I+
kZIr2 --4)+bl[(b-
l)l-2k(r-1)1’
(4.9)
provided thut k( r - 1) f (b - 1 )l. Furthermore, ifthe equality sigrvin (4.9) holds, then INP:y,
+ PSI2
‘+-k2(r2-b)-bkl(r-l)}/{b[k(r-l)-(b-1)0}
is an integer and the given block has {b(p&
+p$42)+k2(r’!
--b)-bkZ(r-
l))/{b[k(r-l)-
(b- l)g}
treatments common with each ofthe remaining (6 -d - 1) blocks. Theorem 11 is a generalization of Theorem I. In fact, when p1 =p2 ( = p9 say), from (4.2) we obtain b(pfqi epiq2) ==pk(b - r) which shows that Theorem 11 yields Theorem I when pl = p2* Note that bounds (4.6) and (4.7) are of the same form as (2.8)
in Theorem I. I-Iowever, (4.6) and (4.7) are derived for a design N with only two nonzero latent roots of NN’ other than rk. IOnthe other hand, (2.8) is derived for a design N with only one con-zero latent root ‘ofNN’ other than rk. Bound (4.9)is represented in terins of parameters of the design I’, latent roots of N N’ and values of 4;s which vary according to the choice of a given block, Because of this fact it may not be easy to evaluate (4.$). On the other hand, bounds (4.6) and (4.7) are conveniently represented in terms of parameters of the design and the maximum latent
@
where p1 ==rk-vUAq, p2=+A1, ~=I,-(l/n)tl,@G,) and ,4@B is the Kronecker product of the matries A =[a,l and B, i.e., A@B= [a$?). Let ai be the number of treatm,ents ccrnmon between a given block of the PBIB design and the i-th group of the ass&&on scheme for i = 1,2,. . ., m. Then from Qz = ( llpz ) N’PN we can obtain
which, from (4.2),yields (4.11) We now consider a regular group divisible PBIB design R19([4]) with parameters u=12,b=32,r=:8,k==3,A,=2,d2=1 andwith32blocks,(1,3,7)(3,5,9)(5,7,11)(1,7, 9) (3,9,11) (1,$11)(2,4,8) (4,6,10) (6,8,12) (2,8,lVj (4,10,12) (2,6,12) (1,3,4) (2,7,9) (3,5,12)(6,9,101(:1,6,8)(2,4,11)(3,7,8)(4,9,12)(5,7,10)(1,10,11)(2,5,6)(3,6, 11)(8, 9,ll) (4,5,8) (4,6,7) (7, i 1,12) (8,10,12) (1,2,12) (2,3,10) (1,5,9), which iniply p1 .-=12 and p2 =6 (pl > p& If the first block (1, 3,7) is chosen as a given block, then y1 = l/l6 and q2 = 1/4from i14.10)and (4.11). For the 13.th block. (1,3,4) we get q, := l/144 a.lldy2 = l3/36. In these cases, the values (by Gauss symbcl) of tl:e right-hand Gde of bounds (4.6) and (4.9) and1the exact values 3.’d in the design are tabulated for etach !(O< I I k = 3) in Tables 1 ad 2, Table 2 appars to show that bound (4,9) satisfactorily rifliits the existing block structure of the design concerning the chosen block (1,3,4). We can produce tables similar to the above for other regular group divisible PBIB designs by * use of (4.10) and @Al). However, they are omitt8d here. Example (ii) Clonsider a 2-ass&ate ‘PBIB design N with parameters u ,-cn(n - 1)/2, b, r, k, Al, A2baseci on a triangular association scheme with association ‘matrices AO,A I and &. Thm it is known (cf. [7]) that the spectral decompocitior of NW is
wh8re
232
--i
0
t
2
3
27 24 1s --.
8 5 3
3 2 0
11
2
3
8 *4 1
3 1 0
;
a.----
(4.6) (4.9) d --
18 14 13
.
Tabie 2 Chosen block (1,3,4) 0
1
.(4.4) (4.9 1 d
18 11 11
27 22 19 -.
and
Let ui be the number of treatments common between a given block of the PBIB design and the i-th row of the association scheme for i = I, 2,. . ., n. We then obtain, from Q1 = (l/pl)N’,PIN, (4.12) which, from (4.2)?yields (4.13) We now consider a triangular PBIB design Tl4([4]) with parameters v = 14 h = 20, r =6,k=3,~,=1,A2=2andvvith20blocks,(1,2,10)(1,5,10)(2~5,lO)(1,8,9)(2,7,8)(2,. 337)(3,7,8~ (1,3,4) f2,6,9’) (2,4,6) (4,6,9) (4, ! 8) (4,810) (1,7,9) (1,6,8) (3,599) (399, 10)(4,5,7) (3,5,6) (6,7, lo), whiichimply pr - 3 and p2 = 6 (p2 r pl). If the first block (1, 2, 10) is chosen as a given block, then q1 -4,45 and q2 - 11,/36from (4.12) andj4.13). For the last block (6,7, lo) we get q1 :r=8/l 5 and q2 = l/12. In these cases, the values (by Gauss’ symbol) of the right hand side Qfbounds (4.7) and (4.9) and the exact values of d in the design are tabulated for each I (0 s I ;l k = 3) in Tables 3 and 4, Table 4 appears to show that bound (4.!3)satisfactorily reflects the existing block structure of the design concerning the chiosen b”aock(6,7, lo), We can also produce tables similar to the above for other triangular PBIB designs of regular type by u.se of (4.12) and (4.131).However, they are not described here.
, .
Upper bsund of block intersection numbers
233
Table4 ChosenMock (6,7,10) I
0
1
2
3
WI
7 4 4
17 1s 15
4 1 0
1 ‘4 0
(49) ‘J
Furthermore, for other PBIB designs based on various association schemes we can produce these tables, but they are omitted here. Acknowledgments This research was partially supported by the Grant 163012 in Aid for General Research (D) of the Ministry of Education, Japan. The authors wish to thank the referee for his useful comments. References lfl] Agrawal, H.L. (1964). On the bounds of the number ofcommon treatments between blocks of certain two associate PBIB designs. Calcutta Statist. Assoc. Bull. 13,76-79. [23 Agrawal,H.L. (1964). Qn the bounds of the number of common treatments between blocks of semiregular group divisible designs, J. Am. Stat&. Aunsoc,59,867~87 1. 133 Agrawal, H.L. (1966). Comparison of the bounds of the number of common treatments between blocks of certainpartially balancedincompleteblock,designs,Ann. Mnth. Stcltist. 37,739- 740. [4] BOSGkc., H. Clarworthy and S.S. Shrikhande (1954). Tables of partially balanced incomplete block designs with two associate classes. North Carolina Agricultural Experimental Bulletin, No. 107. [S] ChaksGarti, MC. (1963). Query. J, Indian Statist. Assoc. 1,230-234. [6] Kageyama, S. (1977). Conditions for a-resolvabilit,y and aftme c+resolvability of incomplete block designs. 1. Japan Statist, Sot. 7, W-25. [7] Kageyama S. and T. Tsuji. Characterization of certain incomplete block designs. J.S.P.I. 1. [8] Kapadia,C.H. (1966). On the bloc% structure of singular group divisible designs. tlnn. Math. Stat%. 37, 1398- 1400, [9] Majindar, RN. (1962). On the parameters and intersection of blocks of balanced inr:omplete block des@ts. ALnn,Math. Statist. 33,12OU-1205. [lo] Raghavarao, D. (1971). Constructions and dombhatorial Problems in Des@ of Experiwzts. John Wiley, New York. ’ B
[ 113. !Shah,S.M. (196.3).On the uppr bound for the number of blocksin balanced incomplete blmk dqipass
having a given numb of treatwants common with a given block. J. Indian S~ist.&soc. 1,2H-220. Sh:nh,S.M. (1964).An upper by ula#for the number of disjoint blocks in cett& I?B!B @signs: Aqw ’ l.L Malk Statist. 35,39&407. Shah, S.M. (1963).Bounds fok the number of commo&eatments b&w&n an+ tie blocks $&r&in PBIB designs.Am. Math Statist. 36,33?-342. Shah, S.M. (1966).On the block structure,of certain #artiaii$ balanced iacomplcta bloik dosip. Ann, Marh. Statist. 3?,1016-1020. Surendlran,P.U. (1968).Cotuman treatments between blocks of certain parti@ balanced . r incomplete bllockdesigns.Ann.Math. Statist. 39.999-1006.