The question I am asking is already asked here.
I want specially the general part, i.e. number of elements of order $r$ in the symmetric group $S_n$, $n\ge 4$. Is there a general rule? Thanks in advance.
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Hint : any permutation $\sigma$ can be written as a product of cycles with disjoint support, and those cylces therefore commute. The order of a cycle is... – Maxime Ramzi Jun 12 '17 at 06:38
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L.c.m of the order of the product cycles. – hiren_garai Jun 12 '17 at 06:42
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This question also appeared at the following MSE link. – Marko Riedel Jun 12 '17 at 20:28
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Oh ! I missed it then – hiren_garai Jun 13 '17 at 01:34
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Every subset $S\subseteq\{1,\ldots,n\}$ with $\operatorname{lcm} S=n$ and map $S\to\Bbb N$ with $\sum_s sf(s)=n$ contributes $\frac{n!}{\prod_s s!^{f(s)}f(s)!}$ to the total count ...
Hagen von Eitzen
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Thanks.,its the same formula for finidng the number of elements in a conjugacy class of a cycle ,right ? – hiren_garai Jun 13 '17 at 01:16
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Well! Will you please cross check the problem : How many elements of order 6 are there in $S_7$ ? , After calculating I found the number to be 1260,as first I found the number of elements in the conjugacy class of a cycle of length 6,then I worked out the number of elements in the conjugacy class of composition of 2cycles(one of length 3 and other of length 2,as the order of the cycle of their composition is lcm of 2 and 3 and hence 6).Now these 2 conjugacy classes are different as their elemnts do not have the same cycle length,so the required number of elements of order 6 is 1260. Right ? – hiren_garai Jun 13 '17 at 01:40