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

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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.

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Is that better? If not, feel free to edit it yourself. – user156822 Jun 13 '14 at 0:35
Yes, much better, thanks. – Nate Eldredge Jun 13 '14 at 0:37
I think, $l$ is $i$, just you have a poor quality copy. – studiosus Jun 13 '14 at 0:49
@studiosus. That makes more sense. – user156822 Jun 13 '14 at 0:57
up vote 5 down vote accepted

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 $$1+1+1+1+2+2+4+5=17\ .$$

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Oh. I see now. $i$ are the $\lambda_l$ that're repeated (for instance the values of i in yours are 1,2,3,4, and 5). Thanks: I knew it was something simple like that -- I was blanking when I looked at it. – user156822 Jun 13 '14 at 0:33

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.

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