The blocks of a balanced ternary design, BTD(V,B; ρ1, ρ2, R;K,Λ), can be partitioned into two sets: the b1 blocks that each contain no repeated elements, and the b2 = B − b1 blocks containing repeated elements. In this note, we address, and answer in some particular cases, the following question. For which partitions of the integer B as b1 + b2 does there exist a BTD(V,B; ρ1, ρ2, R;K,Λ)?

The related genes buttonhead (btd) and Drosophila Sp1 (the Drosophila homologue of the human SP1 gene) encode zinc-finger transcription factors known to play a developmental role in the formation of the Drosophila head segments and the mechanosensory larval organs. We report a novel function of btd and Sp1: they induce the formation and are required for the growth of the ventral imaginal discs....