What does George Andrews mean by i in "Theory of Partitions"?

Question

From the first page of chapter 1 of George Andrews "Theory of Partitions" (Rather ominous place to get stuck):

What do these last two sentences mean? I don't get "where exactly $f_l$ of the $\lambda_j$ are equal to $i$." Can one of you rephrase this for me, because I don't understand what $i$ is.

Answer 1

It just means that the value $i$ is repeated $f_i$ times. For example the notation $(1^42^23^04^15^1)$ means the same as $(1,1,1,1,2,2,4,5)$. Since the parts have to add up to the integer $n$, the sum $$\sum_{i\ge1}f_ii=n$$ in this example is just another (IMHO unnecessarily complicated) way of writing $$1+1+1+1+2+2+4+5=17\ .$$

Answer 2

It means for example that $\lambda = (1^22^33^04^05^1)$ another notation for $\lambda = (1,1,2,2,2,5)$. That is the $f_l$ superscripts tells how many parts of a given size you have.