Elementary formulas for integer partitions

Abstract

In this note we will give various exact formulas for functions on integer partitions including the functions $p(n)$ and $p(n,k)$ of the number of partitions of $n$ and the number of such partitions into exactly $k$ parts respectively. For instance, we shall prove that $$p(n)=\sum _{d|n}\sum _{k=1}^d\sum _{i_0=1}^{\lfloor d/k\rfloor }\sum _{i_1=i_0}^{\lfloor \frac{d-i_0}{k-1}\rfloor }\sum _{i_2=i_1}^{\lfloor \frac{d-i_0-i_1}{k-2}\rfloor }...\sum _{i_{k-3}=i_{k-4}}^{\lfloor \frac{n-i_0-i_1-i_2-...-i_{k-4}}{3}\rfloor }\sum _{c|(d,i_0,i_1,i_2,...,i_{k-3})}\mu (c)(\lfloor \frac{d-i_0-i_1-i_2-...i_{k-3}}{2c}\rfloor -\lfloor \frac{i_{k-3}-1}{c}\rfloor ).$$ Our proofs are elementary.

5 pages

References in corpus (1)

• Combinatorial Identities Involving the Mobius Function