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

Nested Balanced Incomplete Block Designs

If the blocks of a balanced incomplete block design (BIBD) with v treatments and with parameters (v; b1;r;k1) are each partitioned into sub-blocks of size k2, and the b2 =b1k1=k2 sub-blocks themselves constitute a BIBD with parameters (v; b2;r;k2), then the system of blocks, sub-blocks and treatments is, by de4nition, a nested BIBD (NBIBD). Whist tournaments are special types of NBIBD with k1 =2k2= 4. Although NBIBDs were introduced in the statistical literature in 1967 and have subsequently received occasional attention there, they are almost unknown in the combinatorial literature, except in the literature of tournaments, and detailed combinatorial studies of them have been lacking. The present paper therefore reviews and extends mathematical knowledge of NBIBDs. Isomorphism and automorphisms are defined for NBIBDs, and methods of construction are outlined. Some special types of NBIBD are de4ned and illustrated. A first-ever detailed table of NBIBDs with v⩽16, r⩽30 is provided; this table contains many newly discovered NBIBDs. © 2001 Elsevier Science B.V. All rights reserved