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.
Categories: math.NT