Über die Schnittzahlen mehrfach balancierter blockpläne

Abstract

For a finite incidence structure $\text{D}$ with a set $\text{X}$ of blocks let [$\text{X}$] be the number of points common with all blocks contained in $\text{X}$. We define the functions M^(t)(B₁,…; B₁)=Σ_B [B₁, B]…[B₁,B], and, for every partition ϖ = ϖ₁,…,ϖ₁) of t, the function M_ϖ(B₁,…,B₁) = Σ Π_m [B_i | i ϵ R_m], sum over all decompositions {l, …, t} = R₁, ⊃ … ⊃ R_l, |R_m| = ϖ_m. We show: If $\text{D}$ is t-fold balanced, then M^(t) = Σ_ϖ c_ϖM_ϖ, where the, coefficients c_ϖ are linear combinations of the parameters b₁,…,b_t, the constant numbers of blocks through any l,…, t distinct points. Conversely, if the rank of the b × b-matrix ([B, B^∗])_B,B^∗ is equal to the number ν of points and M^(t) is a rational linear combination of the functions M_ϖ, then $\text{D}$ is t-fold balanced.