Association schemes

An association scheme on a finite set Ω is a partition of Ω × Ω into associate classes, satisfying certain rules. They have been invented independently by statisticians and group theorists, and also studied under the name cellular automata.

See the book Association Schemes, or its draft version. There are also some pictures of several different association schemes.

TopicsSome of my publications
Use in partially balanced designs
  • C.-S. Cheng and R. A. Bailey: Optimality of some two-associate-class partially balanced incomplete-block designs. Annals of Statistics 19 (1991), 1667–1671. doi: 10.1214/aos/1176348270 [Maths Reviews 1126346 (92k: 62130)]
  • R. A. Bailey and A. Łacka: Nested row-column designs for near-factorial experiments with two treatment factors and one control treatment. Journal of Statistical Planning and Inference, 165 (2015), 63–77. doi: 10.1016/j.jspi.2015.04.003
  • R. A. Bailey, Peter J. Cameron and Tomas Nilson: Sesqui-arrays, a generalisation of triple arrays. Australasian Journal of Combinatorics, 71 (2018), 427–451. [Maths Reviews 3801275]
  • Association schemes defined by families of partitions (also known as block structures)
  • T. P. Speed and R. A. Bailey: On a class of association schemes derived from lattices of equivalence relations. In Algebraic Structures and Applications (eds. P. Schultz, C. E. Praeger and R. P. Sullivan), Marcel Dekker, New York, 1982, pp. 55–74. [Maths Reviews 0647166 (83f: 06023)]
  • R. A. Bailey: Orthogonal partitions in designed experiments. Designs, Codes and Cryptography 8 (1996), 45–77. [Maths Reviews 1393974 (97g: 62136a) and 1403872 (97g:62136b)]
  • Relationship with group theory
  • P. P. Alejandro, R. A. Bailey and P. J. Cameron: Association schemes and permutation groups. Discrete Mathematics 266 (2003), 47–67. doi:10.1016/S0012-365X(02)00798-7 [Maths Reviews 1991706 (2004c: 05216)]
  • Constructing new association schemes from old ones by crossing, nesting, poset operators, crested products, and more
  • T. P. Speed and R. A. Bailey: On a class of association schemes derived from lattices of equivalence relations. In Algebraic Structures and Applications (eds. P. Schultz, C. E. Praeger and R. P. Sullivan), Marcel Dekker, New York, 1982, pp. 55–74. [Maths Reviews 0647166 (83f: 06023)]
  • R. A. Bailey: Nesting and crossing in design. In Encyclopedia of Statistical Sciences (eds. S. Kotz and N. L. Johnson), J. Wiley, New York, Volume 6, 1985, pp. 181–185.
  • R. A. Bailey: Orthogonal partitions in designed experiments. Designs, Codes and Cryptography 8 (1996), 45–77. [Maths Reviews 1393974 (97g: 62136a) and 1403872 (97g:62136b)]
  • R. A. Bailey and Peter J. Cameron: Crested products of association schemes. Journal of the London Mathematical Society 72 (2005), 1–24. doi: 10.1112/S0024610705006666 [Maths Reviews 2145725 (2006h: 05240)]
  • R. A. Bailey: Generalized wreath products of association schemes. European Journal of Combinatorics 27 (2006), 428–435. doi: 10.1016/j.ejc.2004.11.002 [Maths Reviews 2206477 (2006j: 05125)]
  • More than one association scheme on the same set
  • R. A. Bailey: Suprema and infima of association schemes. Discrete Mathematics 248 (2002), 1–16. [Maths Reviews 1892684 (2003f: 05130)]
  • Designs whose underlying set of experimental units has an association scheme that may not be defined by partitions
  • R. A. Bailey: Designs on association schemes. In Science and Statistics: A Festschrift for Terry Speed (ed. Darlene R. Goldstein), Institute of Mathematical Statistics Lecture Notes-Monograph Series, 40, IMS, Beachwood, Ohio, 2003, pp. 79–102. [Maths Reviews 2004333 (2004k: 62171)]
  • R. A. Bailey: Balanced colourings of strongly regular graphs. Discrete Mathematics 293 (2005), 73–90. doi: 10.1016/j.disc.2004.08.022 [Maths Reviews 2136053 (2006d: 05187)]
  • Strongly regular graphs
  • R. A. Bailey: Balanced colourings of strongly regular graphs. Discrete Mathematics 293 (2005), 73–90. doi: 10.1016/j.disc.2004.08.022 [Maths Reviews 2136053 (2006d: 05187)]
  • R. A. Bailey, Peter J. Cameron, Alexander L. Gavrilyuk and Sergey V. Goryainov: Equitable partitions of Latin-square graphs. Journal of Combinatorial Designs, in press. doi: 10.1002/jcd.21634
  • Page maintained by R. A. Bailey