Some Families of Identities for Integer Partition Function

Abstract

We give a series of recursive identities for the number of partitions with exactly $k$ parts and with constraints on both the minimal difference among the parts and the minimal part. Using these results we demonstrate that the number of partitions of $n$ is equal to the number of partitions of $2n+d\left(\genfrac{}{}{0ex}{}{n}{2}\right)$ with $n$ $d$-distant parts. We also provide a direct proof for this identity. This work is the result of our aim at finding a bijective proof for Rogers-Ramanujan identities.

9 pages

References in corpus (1)

• Formulae for the number of partitions of n into at most m parts (using the quasi-polynomial ansatz)