Symmetric Groups and Commutativity I just finished my homework which involved, among many things, the following question:
Let $S_{3}$ be the symmetric group $\{1,2,3\}$. Determine the number of elements that commute with (23). 
Now, solving this was unproblematic - for those interested; the answer is 2. However, it got me thinking whether or not there is a general solution to this type of question.Thus, my question is:
Let $S_{n}$ be the symmetric group $\{1,\dots,n\}$. Determine the number of elements in $S_{n}$ that commute with $(ij)$ where $1\leq i,j \leq n $. 

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*Let $\pi, \phi\in S_{\Omega}$. Then $\pi, \phi$ are disjoint if $\pi$ moves $\omega\in \Omega$ then $\phi$ doesn't move $\omega$.


For example, $(2,3)$ and $(4,5)$ in $S_6$ are disjoint. indeed, $\{2,3\}\cap\{4,5\}=\emptyset$.
Theorem: If $\pi, \phi\in S_{\Omega}$ are disjoint then $\pi\phi=\phi\pi$.
