Counting the number of 2-cycles, 3-cycles and 4-cycles in $S_4$

Question

I know that in $S_4$ there are five different cycle types: double transpositions, 2-cycles, 3-cycles, 4-cycles and 1-cycle (i.e. identity).

My question is, does there exist a combinatorial formula or similar for counting the number of, for example, the number of 3-cycles in $S_4$?

Answer 1

From Roman's "Fundamentals of Group Theory: An Advanced Approach", page 200, we have:

Theorem 6.12:

1. The number of cycles of length $k$ in $S_n$ is $$\binom{n}{k}(k-1)!=\frac{n!}{k(n-k)!}.$$

2. The number of permutations in $S_n$ whose cycle structure consists of $r_i$ cycles of length $k_i$, for $i=1,\dots, m$ is $$\frac{n!}{r_1!\dots r_m!k_1^{r_1}\dots k_m^{r_m}}.$$

Answer 2

The set $\{1,2,3,4\}$ has $\binom{4}{3}=4$ subsets of cardinality 3, e.g. $\{1,2,3\}$. Each of them gives $(3-1)!=2$ different 3-cycles: $(1,2,3)$ and $(1,3,2)$, once you fix the first entry there are $2!$ possibilities for the rest.

In total there are $4\cdot 2=8$ different 3-cycles in $S_4$.

In general there are $\binom nk\cdot(k-1)!$ differen $k$-cycles in $S_n$.