Exercise 1.3.17 (Number of permutations which are the product of two disjoint $2$-cycles)

Proof. Let $ℐ([[1,4]],[[1,n]])$ be the set of injections of $[[1,4]]$ to $[[1,n]]$

We write $\left(\begin{array}{cccc}\hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill & \hfill 4\hfill \\ \hfill a\hfill & \hfill b\hfill & \hfill c\hfill & \hfill d\hfill \end{array}\right)$ the injection $(1\mapsto a,2\mapsto b,3\mapsto c,4\mapsto d)$, where $a,b,c,d\in [[1,n]]$ are distinct integers. Let $𝒫$ denote the set of product of two disjoint $2$-cycles.

By Exercise 16,

(We choose $a$ among $n$ elements, $b$ among $n-1$ elements, $c$ among $n-2$ elements, and $d$ among $n-3$ elements.)

Consider the map

which sends the injection $(1\mapsto a,2\mapsto b,3\mapsto c,4\mapsto d)$ on $(ab)(cd)$. Since $a,b,c,d$ are distinct, $(ab)(cd)$ is a product of two disjoint $2$-cycles, so the map $\phi$ is well defined.

Let $(ab)(cd)$ be any permutation in $𝒫$. Then $(ab),(cd)$ commute (and $(ab)=(ba)$), thus

(but $(ab)(cd)\ne (ac)(bd),\dots$). Every permutation $(ab)(cd)$ is the image of exactly $8$ injections.

There is no other preimage: indeed if $(ab)(cd)=(a^{\prime }{b}^{\prime })(c^{\prime }{d}^{\prime })\in 𝒫$, then ( $\{a,b\}=\{a^{\prime },b^{\prime }\}$ and $\{c,d\}=\{c^{\prime },d^{\prime }\}$), or ( $\{a,b\}=\{c^{\prime },d^{\prime }\}$ and $\{c,d\}=\{a^{\prime },b^{\prime }\}$), so there is no other solution.

This shows that each permutation $\sigma \in 𝒫$ has exactly $8$ preimages: for every cycle $\sigma \in 𝒫$,

Since

we obtain

so

The number of permutations in $S_n$ which are the product of two disjoint $2$-cycles is $n(n-1)(n-2)(n-3)∕8$. □