Find the circle types with three cycles of a set

Question

This question involves the permutation of the set $N=\{1,2,3,4,5,6,7\}.$ What are the cycle types with exactly three cycles? And, for each of these types, how many cycles are there?

Answer 1

So you need to get partition of 7 as a sum of three integers. I am assuming 1 cycles are allowed.

7= 1+1+5=1+2+4=1+3+3=2+1+4=2+2+3.I have written in ascending order.

Now you can use counting principles for example for (2, 2,3) case it is ${7 \choose 2} \times { 5\choose 2} \times (3-1)!/2$ . Hope it helps. The principal goes as follows: for a r cycle choose r integers from $\{1,..., n\}$ in ${n \choose r}$ ways then multiply it with $(r-1)!$ to get all possible cycles coming out of those r choices of integers then do the similar thing with rest of n-r elements and if you have same r appearing $ k_r$ number of times in the partition divide it by $k_r! $ for each of those cases.

The complete formula will look like this: Suppose for a particular cycle type in $S_n$ $m_1, m_2,.., m_s$ be the distinct integers appearing in the list and $m_i$ appears $k_i$ times then the number of elements satisfying that cycle type is $\frac {n! }{\prod k_i! m_i^{k_i}}$.