# Cycle type of a permutation in $S_n$ and its relation to partition of $n$ and its Young diagram

I know that it's possible to assign to each permutation its cycle type. I found two definitions of the cycle type and its relation to a partition of $$n$$:

First definition

Given $$\sigma \in S_n$$ written as product of $$l$$ cycles of lengths $$(k_1,\dots,k_l)$$, the cycle type is just $$(k_1,\dots,k_l)$$. Then by ordering the product so that $$k_1 \geq k_2 \geq \dots \geq k_l$$ and seeing that $$\sum_i k_i = n$$, one obtains that $$(k_1,\dots,k_l)$$ is a partition of $$n$$.

Second definition

Given $$\sigma \in S_n$$ the cycle type is the list $$(w_1,...,w_n)$$ where $$w_i$$ is the number of $$i$$-cycles in the product. Then it's possible to build a partition $$\lambda = (\lambda_1,\dots,\lambda_n)$$ of $$n$$ in the following manner:

• $$\lambda_1 = \sum_{i=1}^{n} w_i$$
• $$\lambda_2 = \sum_{i=2}^{n} w_i$$
• $$\dots$$
• $$\lambda_n = w_n$$

since $$\lambda_1 \geq \lambda_2 \geq \dots \geq \lambda_n$$ and $$\sum_{i=1}^{n} \lambda_i = n$$ we know that $$\lambda$$ is indeed a partition of $$n$$.

Young Diagram

I know that we can show a partition graphically using a Young diagram where each row, starting from the top (in the English style), contains a number of boxes that is equal to the i-th number listed in the partition.

So, by the previous definitions, a Young diagram would have $$k_1$$ or $$\lambda_1$$ in its first row, then $$k_2$$ or $$\lambda_2$$ in its second row, and so on.

Question I'm getting confused because when I apply these two definitions, given a $$\sigma \in S_n$$ I get two different partitions and hence two different Young diagrams.

For example let $$\sigma = (123)(45) \in S_5$$. If we use the first definition, the cycle type would be $$(3,2)$$ and the corresponding Young diagram would have $$3$$ boxes in the first row and $$2$$ boxes in the second.

However, if we use the second definition the cycle type ($$w$$) would be $$(0,1,1,0,0)$$ while the partition ($$\lambda$$) would be $$(2, 2, 1, 0, 0)$$ so that the Young diagram would have $$2$$ boxes in the first row, $$2$$ boxes in the second row and $$1$$ box in the third row.

Which one is the correct definition?

Thanks.