Balanced 6x6 designs for 4 equally replicated treatments

This paper considers 6 x 6 row-and-column designs in which (i) each of 4 treatments is replicated 9 times, (ii) each treatment occurs either once or twice in each row and each column, (iii) exactly 2 treatments are duplicated in each row and each column, and (iv) each of the 6 pairs of treatments is duplicated in exactly one row and exactly one column. These designs are balanced in a statistical as well as combinatorial sense and have been of statistical interest for more than 40 years. But they have not hitherto been enumerated. A backtracking search process [6] has now been used to enumerate the designs that are inequivalent under independent row permutations, column permutations, relabelling of the treatments and transposition about the main diagonal; just over half-a-million inequivalent designs were found. With attention restricted to designs symmetrical about the main diagonal, 920 designs were found, including only 2 with automorphism group of size 12 (the maximum possible). Some balanced superimpositions of one of the designs of another are presented.