On an Identity Relating to Partitions and Repetitions of Parts

Extract

This note is concerned with a simple but rather surprising identity which emerged unexpectedly from the work of one of the authors on the characterisation of characters. Consider, for example, the seven partitions of 5. These are

(1) 5, 4 1, 3 2, 3 1², 2² 1, 2 1³, 1₅

and with each of these we can associate a product of factorials of the numbers of repetitions, respectively

(2) 1!, (1!)(1!), (1!)(1!), (1!)(2!), (2!)(1!), (1!)(3!), 5!

It is then seen that the product of all the numbers occurring in (1) coincides with that of all the numbers in (2).

Generally, for any particular natural number n, the partitions can be written in the form

1^a₁ 2^a₂ 3^a₃…,

in which a_k is the frequency of repetition of the part k, and are enumerated by the distinct sets {α} = {a₁, a₂, …,} with a_k ≧ 0 and Σka_k = n.