A positive integer can be partitioned, for example, the number 7 can be partitioned into the following:
$7=7$
$ 7=6+1$ , $ \ \ 7=5+2$,$ \ \ 7=4+3$
$ \ \ 7=4+2+1$,$ \ \ 7=3+3+1$,$ \ \ 7=3+2+2$,
$ \ \ 7=2+2+2+1$,...
I consider $n_k$ as the number of times that a number is used. For example, in partitioning $ 7 = 3 + 2 + 2$, we have $n_2 = 2$ and $ n_3 = 1$
suppose $K$ as largest element in every partiotioning , For example, in partitioning $ 7 = 3 + 2 + 2$ , $K$ is $3$ , and in the partitioning $ 7 = 5 + 2 $ , we have $K=5$ .
let $\dbinom{1}{2}=0$ ,I want to know from all the above combinations Which one have smaller $P=\sum_{k=1}^K \dbinom{k}{2} n_k$ .(For example, in partitioning $ 7 = 3 + 2 + 2$ , this value is $P=\dbinom{3}{2} +2* \dbinom{2}{2} = 5 $)
I mean, which combination(partiotioning ) has the smallest value of $P=\sum_{k=2}^K \dbinom{k}{2} n_k$ ?
thanks