Partition Identities

Citations

Previous

Up

Next

Article

From References: 4 From Reviews: 4

MR0306105 (46 #5232) 10A45 Andrews, George E. Partition identities. Advances in Math. 9, 10–51 (1972). The author considers the general study of partition identities and, in particular, attempts to bring some unity to the subject by introducing the point of view of lattice theory. Denote by S the set of all sequences {fi } of nonnegative integers having only finitely many nonzero terms. The author observes that S forms a distributive lattice underPthe partial ordering {fi } ≤ {gi } provided fi ≤ gi for each i. Define the function σ by σ({fi }) = ∞ i=1 fi · i; thus {fi } can be regarded as defining a partition of σ({fi }), with fi giving the multiplicity of the part i in the partition. Call a subset C of S a partition ideal if it is a semi-ideal in the lattice S, i.e., if whenever a ∈ C, x ∈ S and x ≤ a, it follows that x ∈ C. Define p(C; n) to be the cardinality of the set {Π|Π ∈ C, σ(Π) = n} and say that C1 ∼P T C2 if for each n ≥ 0, p(C1 ; n) = p(C2 ; n). Any relation of this type clearly yields a partition identity. Almost all known partition identities can be framed in these terms. The author poses the problem of describing fully the equivalence classes in S under the equivalence relation ∼P T . This problem is no doubt inaccessible at present, but the author is able to give some useful information about the structure of the equivalence classes. In particular, many of them are infinite. In addition to presenting the above approach, the author gives a brief survey of the subject of partition identities, detailed examples of the use of q-difference equations and more elementary methods to derive identities, and a large bibliography. This paper should be of great value to those who wish to work in the field. Reviewed by S. A. Burr c Copyright American Mathematical Society 1973, 2007