Identically Distributed Pairs of Partition Statistics

We show that many theorems which assert that two kinds of partitions of the same integer n are equinumerous are actually special cases of a much stronger form of equality. We show that in fact there correspond partition statistics X and Y that have identical distribution functions. The method is an extension of the principle of sieve-equivalence, and it yields simple criteria under which we can infer this identity of distribution functions. Let P(n) be the set of partitions of the integer n, p(n) = |P(n)|, and let P = ∪n≥1P(n) be the set of all partitions of all positive integers. A partition statistic X is a nonnegative-integervalued function defined on P . Two partition statistics X and Y are identically distributed if ∀n ≥ 1,∀j ≥ 0 : Probn(X = j) = Probn(Y = j), where Probn(X = j) = def |{π ∈ P(n) : X(π) = j}| p(n) . There are a number of classical theorems of the form “The number of partitions of n that have no . . . is the same as the number that have no . . . .” The purpose of this note is to remark that frequently in such cases there is an underlying pair of identically distributed partition statistics. For example, Euler’s famous theorem that the number of partitions of n with distinct parts is the same as the number with odd parts can be strengthened to the following. Theorem 1 The partition statistics X(π) = number of even part sizes that occur in π, and Y (π) = number of repeated part sizes that occur in π, are identically distributed. That is, for every n and j, the number of partitions of n in which exactly j different sizes of parts are repeated is the same as the number in which exactly j different sizes of parts are even, and Euler’s original theorem is the case j = 0. Thus a great deal more is true than is stated in the original theorem.