Notation for parts of an integer partition

Question

If I have an integer partition of $n \in \mathbb{N}$

$$n=j_1 + 2 j_2 +3 j_3 + \dots + ij_i$$

and say the example is

$$10 = 1(2) + 2(0) + 3(1) + 4(0) + 5(1) = 1 +1 +3 + 5$$

what is the notation for the parts of the partition (the "summands"), here $\{1,3,5\}$? Is there a standard way or should I just define it like this ?

Answer 1

You could write the series as a summation like this- $$n={\sum_{k=1}^i}\,k{\cdot}j_k$$

Answer 2

With $m_{i}=m_{i}(\lambda)$ we denote the number of parts of the partition $\lambda$ equal to $i$. For the partition $\lambda$ of $n$ we use the notation: $\lambda=(1^{m_{1}}, 2^{m_{2}},\ldots)$.

For example, we can rewrite the partition $(4,3,3,2,1,1,1)$ of $n=15$ as $(1^{3},2,3^{2},4)$. This is done purely for purposes of brevity.