Some Formulae For Partitions Into Squares

. We consider the new problem of determining the number of partitions of a number into a fixed number k of squares, and find explicit formulae in the cases k = 2, 3, 4. Classification: 11P81, 05A17 Keywords: partitions, squares. 1. Introduction and statement of results We consider the problem of determining the number of partitions of a number into a fixed number of squares. This problem is distinct from, but not unrelated to, the corresponding classical problem of determining the number of representations of a number as the sum of a fixed number of squares. Whereas the classical problem has received an enormous amount of attention over the years, see Dickson [3, Vol. II, Chapters VI--IX], it appears that this (restricted) partition problem has not previously been investigated. Hardy and Ramanujan [4, p.305] and Baxter [1] have looked at the problem of partitions of a number into (an unrestricted number of) squares, which finds application in statistical mechanics. To see why one m..

Abstract. SOME FORMULAE FOR PARTITIONS INTO SQUARES

We consider the new problem of determining the number of partitions of a number into afixednumberkofsquares, and find explicit formulae in the cases k =2, 3, 4. Classification: 11P81, 05A1