On the polynomial representation for the number of partitions with fixed length

Abstract

In this paper, it is shown that the number M(n, k) of partitions of a nonnegative integer n with k parts can be described by a set of (k) over tilde polynomials of degree k-1 in Q((k) over tilde), where (k) over tilde denotes the least common multiple of the k integers 1, 2, . . . , k and Q((k) over tilde) denotes the quotient of n when divided by (k) over tilde. In addition, the sets of the (k) over tilde polynomials are obtained and shown explicitly for k = 3, 4, 5, and 6.