Size-minimal combinatorial designs of staircase type

Abstract

Given a positive integer $n$ and a partitioning $n=r_1{s}_1+\cdots +r_t{s}_t$, $t,r_i,s_i$ positive integers, such that $r_1>\cdots >r_t$ (for $t\ge 2$), we can write $n$ symbols $1,\dots ,n$ in the form of a staircase matrix having $r_1$ rows where first $r_1-r_2$ rows have $x_1$ columns, next $r_2-r_3$ rows have $t_1+t_2$ columns, etc., and finally last $r_t$ rows have $t_1+\cdots +t_k$ columns. Then we can construct a~design having $r_1+s_1+\cdots +s_t$ sets by taking all $r_1$ rows and $s_1+\cdots +s_t$ columns of this staircase matrix. Such designs have exactly two replications of each symbol and various cardinalities for the sets constituting the design. The minimum size of combinatorial designs of staircase type is found.