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 $\widetilde {k}$ polynomials of degree $k-1$ in $Q_{\widetilde {k}}$, where $\widetilde {k}$ denotes the least common multiple of the $k$ integers $1, 2, \cdots , k$ and $Q_{\widetilde {k}}$ denotes the quotient of $n$ when divided by $\widetilde {k}$. In addition, the sets of the $\widetilde {k}$ polynomials are obtained and shown explicitly for $k=3, 4, 5,$ and $6$.