NOTE. A Note on Fisher's Inequality

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1 journal of cominatorial theory, Series A 77, (1997) article no. TA NOTE A Note on Fisher's Inequality Douglas R. Woodall Department of Mathematics, University of Nottingham, Nottingham NG7 2RD, Engl Communicated y Laci Baai Received May 3, 1995 A new proof is given of the nonuniform version of Fisher's inequality, first proved y Majumdar. The proof is ``elementary,'' in the sense of eing purely cominatorial not using ideas from linear algera. However, no nonalgeraic proof of the n-dimensional analogue of this result (Theorem 3 herein) seems to e known Academic Press A design D consists of a family B 1,..., B of suets, called locks, ofa finite set S=[P 1,..., P v ] whose elements are called points or varieties. D is alanced or *-linked if every pair of points is contained in exactly * locks. If, in addition, *>0 no lock contains all the points, then D is nontrivial, if every lock has the same cardinality k then D is a alanced incomplete-lock design or BIBD. Fisher [5] proved that if D is a BIBD, then v. Bose [3] gave a neat short proof of this result using a determinant. Majumdar [8] provided an easy modification of Bose's method that extends the result to aritrary non-trivial *-linked designs, which one can think of as a nonuniform version of Fisher's inequality. (The case *=1 of Majumdar's result had een proved earlier y de Bruijn Erdo s[4].) My attention has recently een drawn to the statement of Baai [1] that no proof of Majumdar's inequality appears to e known that does not use some form of linear algera trick. Accepting the challenge, I offer the proof elow (Theorem 1). Fisher's proof relies on the fact that the variance of the quantities B i & B j (i{j), eing a sum of squares, is nonnegative, his proof shows that (when D is a BIBD) these quantities are all equal if only if =v. The key to the proof elow (which involved a fair amount of hindsight) was to discover a similar sum of squares in the nonuniform case, which is equal to zero if only if =v Copyright 1997 y Academic Press All rights of reproduction in any form reserved.

2 172 NOTE Ryser [11] Woodall [12] considered the case of equality in Majumdar's theorem independently made the same conjecture, which is still open, is usually referred to in Ryser's terminology as the *-design conjecture. (A*-design is what one gets y taking a non-trivial *-linked design with =v that is not a BIBD, dualizing it, that is, interchanging the ro^ les of points locks. Most recent authors have followed Ryser in writing in this dual terminology, ut I shall keep to the original formulation of Fisher, Bose Majumdar.) Sadly if unsurprisingly, the following proof of Theorem 1 seems to give no extra information aout the cases of equality that might help in proving the *-design conjecture; as we shall see in Theorem 2, the equations otained seem identical to those otained y the use of linear algera in [11] [12]. Theorem 1. If D is a non-trivial *-linked design with v points locks, then v. Proof. For each point P in S, let r e the numer of locks containing P let \ = (r &*) &1, called the residue of P. As in [12], we define R$ = \, P #S R i = \, P # B i R ij = \ P # B i & B j (1) for i, j # [1,..., ], R =R$+* &1. (2) (Note that R ii =R i.) Since r \ =1+*\, counting the numer of times \ is involved in each sum, (1) gives In a similar way, R 2 i = \ R i = P # S r \ =v+*r$ (3) R ij = P # B j r \ = B j +*R j. (4) \ 2 + P # B i = r \ 2 + {; P, P ; # B i \ \ ;+ {; *\ \ ; = (r &*)\ 2 +* \ 2 \ + =R$+*R$ 2 =*RR$, (5)

3 NOTE 173 R 2 = ij r \ 2 + *\ \ ; =R j +*R 2 j (6) P # B j P, P ; # B j {; By (5), (6) (7), since 0 i{j = R ij (R i &R ij )= *\ \ ; =*R j (R$&R j ). (7) P # B j P ; B j (RR ij &R i R j ) 2 (8) [(R 2 &2RR j )R 2 ij &2RR j R ij (R i &R ij )+R 2 i R2 j ]&(RR j&r j R j ) 2 =(R 2 &2RR j )(R j +*R 2 j )&2*RR2 j (R$&R j)+*rr$r 2 j &(RR j&r 2 j )2 =R j (R&R j )(R&RR j +R 2 j ) (9) (R 2 &2RR j )(R j +*R 2 j )=R2 R j +*RR$R 2 j &RR2 j &2*RR3 j y (2). Since R j >0 R&R j >0, if follows from (9) that R&RR j +R 2 j 0 (10) for each j (1 j). Summing (10) over all j using (3) (5) we otain R&R(v+*R$)+*RR$=R(&v)0. Since R>0, this gives v as required. The aove proof gives exactly the same information as the algeraic proof [11, 12] aout the cases of equality in Theorem 1 Theorem 2. If D is a non-trivial *-linked design with v points locks, where =v, D is not a BIBD, then D has locks of exactly two distinct sizes k 1 k 2, where k 1 +k 2 =v+1. Moreover, if we define K S 1 = v&1 k 2 &1 S 2 = v&1 k 1 &1, (11)

4 174 NOTE then R i =S 1 or S 2 according as B i =k 1 or k 2, S 1 &1 if B i = B j =k 1, R ij ={S 2 &1 if B i = B j =k 2, (12) 1 if B i { B j. Proof. If =v, then all the inequalities in the proof of Theorem 1 are equalities, (10) (8) give for each j R&RR j +R 2 j =0 (13) RR ij &R i R j =0 (14) whenever i{j (i, j # [1,..., ]). By (13), there are at most two possile values for R j, if (anticipating somewhat) we denote these y S 1 S 2 then (13) (14) can e comined as S 1 +S 2 =S 1 S 2 =R. (15) R($ ij &R ij )+R i R j =0 (16) where $ ij is the Kronecker delta, summing (16) over all i, using (2), (3) (4), we find that is, where R(1& B j &*R j )+R j (v&1+*r)=0, B j =1+R j (v&1)r=k 1 or k 2, k 1 =1+S 1 (v&1)r, k 2 =1+S 2 (v&1)r (17) k 1 +k 2 =2+(S 1 +S 2 )(v&1)r=2+v&1=v+1

5 NOTE 175 y (15). Also, (15) (17) give (11). Finally, if B i = B j =k 1, then (14) (13) give R ij = S 2 1 R =RS 1&R =S R 1 &1, the rest of (12) follows similarly from (14) (15). In [13] I gave a short algeraic proof of the following n-dimensional analogue of Theorem 1. Theorem 3. Suppose we are given a finite set S=[P 1,..., P v ], positive integers n * 2,..., * n, n families of proper susets of S called t-locks (t=1, 2,..., n), such that (i) the 1-locks are precisely the singletons [P ]ofs, (ii) for each t2, for each (t&1)-lock B each P in S"B there are exactly * t t-locks containing B _ [P ]. Then for each t (1tn), the numer t of t-locks satisfies t v. The case n=2 of this result is Theorem 1. The case in which all the * i s equal 1 is also well known, eing the cominatorial analogue of Motzkin's hyperplane inequality [10]. This asserts that if v points in a Euclidean or projective space do not all lie in the same hyperplane (=affine or projective suspace of codimension 1), then they determine at least v distinct hyperplanes. The cominatorial generalization of this is the analogous statements aout matroids (that the numer of hyperplanes is at least as large as the numer of atoms); it follows easily from Motzkin's work (see Mason [9]), was proved directly independently y Basterfield Kelly [2], Greene [6] Heron [7]. Although this special case can e proved without using ideas from linear algera, I do not know of any nonalgeraic proof of Theorem 3 itself. K REFERENCES 1. L. Baai, On the nonuniform Fisher inequality, Discrete Math. 66 (1987), J. G. Basterfield L. M. Kelly, A characterization of sets of n points which determine n hyperplanes, Proc. Camridge Philos. Soc. 64 (1968), R. C. Bose, A note on Fisher's inequality for alanced incomplete lock designs, Ann. Math. Statist. 20 (1949), N. G. de Bruijn P. Erdo s, On a cominatorial prolem, Proc. Konink. Nederl. Akad. Wetensch. Ser. A 51 (1948), ; Indag. Math. 10 (1948), R. A. Fisher, An examination of the different possile solutions of a prolem in incomplete locks, Ann. Eugenics 10 (1940), 5275.

6 176 NOTE 6. C. Greene, A rank inequality for finite geometric lattices, J. Comin. Theory 9 (1970), A. P. Heron, A property of the hyperplanes of a matroid an extension of Dilworth's theorem, J. Math. Anal. Appl. 42 (1973), K. M. Majumdar, On some theorems in cominatorics relating to incomplete lock designs, Ann. Math. Statist. 24 (1953), J. H. Mason, Matroids unimodal conjectures Motzkin's theorem, in ``Cominatorics'' (D. J. A. Welsh D. R. Woodall, Eds.), pp , Inst. Mathematics Applications, Southend-on-Sea, Essex, Engl, Th. Motzkin, The lines planes connecting the points of a finite set, Trans. Amer. Math. Soc. 70 (1951), H. J. Ryser, An extension of a theorem of de Bruijn Erdo s on cominatorial designs, J. Algera 10 (1968), D. R. Woodall, Square *-linked designs, Proc. London Math. Soc. (3) 20 (1970), D. R. Woodall, The inequality v, in ``Proc. Fifth British Cominatorics Conference, 1975,'' Congress. Numer. 15 (1975),