Perfect Graeco-Latin balanced incomplete block designs (pergolas)

A PERfect GraecO-LAtin balanced incomplete block design (PERGOLA) is a block design for two sets of treatments, where (a) each set is arranged relative to the blocks in a balanced incomplete block design (BIBD), (b) each set is arranged relative to the other in a symmetric BIBD, and (c) the overall arrangement is such that there is adjusted orthogonality between the two sets. The currently small literature of pergolas is reviewed, and the topic is shown to be rich in combinatorial interest and unsolved problems. Isomorphism, automorphisms and duality are defined for pergolas, and matters of existence are discussed. A first-ever extensive table of pergolas with r less than or equal to 20 is presented. For each of many of the 66 parameter-sets covered, the Table gives a selection of non-isomorphic pergolas, perhaps based on a selection of non-isomorphic BIBDs for that parameter-set.