UDC 519.1

MATHEMATICS

A. Ya. Petrenyuk

ON THE EXISTENCE OF WEIGHTED FINITE INCIDENCE STRUCTURES

(Presented by Academician P. S. Aleksandrov on 21 I 1970)

The article introduces the concept of a weight function of a finite incidence structure; with the aid of this concept, necessary conditions are found for the existence of \(l\)-weighted structures, defined below.

Let \(v\) be a natural number, \(v \ge 4\), and let \(E\) be a set consisting of \(v\) elements. The symbol \(P(E)\) denotes the collection of subsets of the set \(E\). Let \(n(\beta)\) be a function defined on \(P(E)\) and taking values in the set \(Z^+\) of nonnegative integers. This function defines on \(E\) a (finite) incidence structure (briefly: a structure), containing the subset \(\beta \subseteq E\) exactly \(n(\beta)\) times. The elements of the set \(P(E)\) that enter the structure are called blocks, and the function \(n(\beta)\) is called the function of block multiplicities in the structure. If \(H\) is a finite set, then \(|H|\) often denotes the number of elements in \(H\); the number \(|\beta|\) is called the volume of the block \(\beta\).

Each structure defines on \(P(E)\) a function

\[ \rho(X)=\sum_{\{\beta:X\subseteq\beta\}} n(\beta), \tag{1} \]

which we shall call the weight function of the structure.

Let \(K=\{k_1,\ldots,k_s\}\) be a set of natural numbers; let \(l\) be a natural number, \(2\le l<k_1<\cdots<k_s<v\); let \(P_l(E)=\{L:L\subset E,\ |L|=l\}\), and suppose that on \(P_l(E)\) a function \(\lambda(L)\) with values in \(Z^+\) is given. A structure for each block \(\beta\) of which the inclusion \(|\beta|\in K\) holds and in which for every \(L\in P_l(E)\) the equality

\[ \rho(L)=\lambda(L), \tag{2} \]

holds is called an \(l\)-weighted structure of type \(C(K,l,\lambda(L),E)\). The substructure \(A_i\) of an \(l\)-weighted structure \(A\), defined by the block-multiplicity function

\[ n_i(\beta)= \begin{cases} n(\beta), & \text{if } |\beta|=k_i,\\ 0 & \text{otherwise,} \end{cases} \]

is called the \(i\)-th equiblock component of the structure \(A\); the weight function of \(A_i\) is denoted by \(\rho_i(X)\).

Counting in two different ways the number of occurrences of subsets \(D\) such that \(T\subseteq D\subset E\), in the \(i\)-th equiblock component, we obtain that the equality

\[ \binom{k_i-t}{d-t}\rho_i(T) = \sum_{\substack{H\subseteq E\setminus T\\ |H|=d-t}} \rho_i(T\cup H), \tag{3} \]

is valid, where \(d\) is a natural number, \(0\le |T|=t<d\le k_i,\ T\subset E\).

Putting \(d=l\) in (3) and summing the equalities (3) with respect to the index \(i\), taking into account (2) and the obvious formula
\[ \rho(X)=\sum_{i=1}^{s}\rho_i(X), \]
we obtain the following theorem.

Theorem 1. Suppose there is an \(l\)-weighted structure of type \(C(K,l,\lambda(L),E)\). Then the relation
\[ \sum_{i=1}^{s}\binom{k_i-t}{l-t}\rho_i(T) = \sum_{\substack{H\subseteq E\setminus T\\ |H|=l-t}} \lambda(T\cup H) \tag{4} \]
holds for every \(T,\ T\subset E,\ 0\le |T|=t<l\).

We shall denote by
\[ d\left[\binom{k_i-t}{l-t}\right]_{i=1}^{s} \]
the greatest common divisor of the integers
\[ \binom{k_1-t}{l-t},\ldots,\binom{k_s-t}{l-t}. \]

Theorem 2. For the existence of an \(l\)-weighted structure of type \(C(K,l,\lambda(L),E)\), it is necessary that the conditions
\[ \sum \lambda(T\cup H)\bigg/ d\left[\binom{k_i-t}{l-t}\right]_{i=1}^{s} =\text{an integer} \tag{5} \]
hold for all \(T,\ T\subset E,\ 0\le |T|=t<l\). The summation in (5) is taken over all \(H,\ H\subset E\setminus T,\ |H|=l-t\).

Condition (5) follows directly from (4).

In the case when the function \(\lambda(L)\) is identically equal to the natural number \(\lambda\), an \(l\)-weighted structure of type \(C(K,l,\lambda,E)\) is called \(l\)-balanced. For \(s=1\) the latter is a tactical configuration \((^{1,3})\), which for \(l=2\) is called a balanced incomplete block design \((^2)\). Condition (5) for the case of \(l\)-balanced structures takes the form
\[ \lambda\binom{v-t}{l-t}\bigg/ d\left[\binom{k_i-t}{l-t}\right]_{i=1}^{s} =\text{an integer},\quad t=0,1,\ldots,l-1. \tag{6} \]

Relation (4) for \(\lambda(L)\equiv\lambda,\ l=2\) and \(t=0\) occurs in \((^2)\). Conditions (6) for \(s=1\) turn into the known existence conditions for tactical configurations \((^1)\).

Faculty of Mechanics and Mathematics
M. V. Lomonosov Moscow State University

Received
16 I 1970

References

1. H. Hanani, Canad. J. Math., 15, No. 4, 702 (1963).
2. R. C. Bose, Canad. J. Math., 12, No. 2, 177 (1960).
3. A. Ya. Petrenyuk, Mat. Zametki, 4, No. 4, 417 (1968).