Determine $\left|\left\{g\in S_4 : g^2=e\right\}\right|$

Question

Determine $\left|\left\{\in S_4 : g^2=e\right\}\right|$.

The answer of this question should be $9$? because $g=(12)(34), (13)(24), (14)(23), (12), (13), (14), (23), (24), (34)$.

Answer 1

The question is about the number of involutions in the symmetric group on $n$ symbols. This is also linked to telephone lines!, as it equals the number of ways $T_n$ a telephone service can connect two subscribers, given that the service has $n$ subscribers. Thanks to this wikipedia article, one has $$ T_n = \sum_{k=0}^{\lfloor n/2\rfloor}{n\choose 2k}\frac{(2k)!}{k!2^k} = |\mathrm{He}_n(i)|, $$ where $i := \sqrt{-1}$ and $\mathrm{He}_n$ is the (probabilist's) $n$th Hermite polynomial. For example,

• $\mathrm{He}_0(x) = 1$,
• $\mathrm{He}_1(x) = x$,
• $\mathrm{He}_2(x) = x^2-1$,
• $\mathrm{He}_3(x) = x^3 - 3x$,
• $\mathrm{He}_4(x) = x^4 - xz^2 + 3$.

Now, back to the original question, it is then an easy computation that $$ T_4 = |\mathrm{He}_n(i)| = |i^4 - 6i^2 +3| = 1 + 6 + 3 = 10. $$