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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.

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I have never seen the second definition to be honest. For me the cycle type is given by the first one and at least for me that is the classical one. Most often one defines the cycle type to see that the cycle type determines the conjugacy class, i.e. two permutations have the same cycle type iff they are conjugate and for me that is more intuitive via the first one. There is a way to convert them into each other though by conjugation.

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  • $\begingroup$ Thank you ThorWittich! $\endgroup$ – Pietro Bernardi Jun 22 '19 at 8:29

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