= Symmetric-group involution class-size collision
{title2=$n!/[2^kk!(n-2k)!]=\binom n2\iff k=1\text{ or }(n,k)=(6,3)$}
A <permutation> of order two is a product of $k$ disjoint <transpositions>. Its <conjugacy class> has the displayed size. For $k\geq2$, equality with the <transposition> class requires $(n-2)!/(n-2k)!=2^{k-1}k!$. The left side grows strictly with $n$. At $n=2k$, its ratio to the right side is one only for $k=3$; at $k=2$ it jumps from below to above one between $n=4$ and $n=5$, and for $k\geq4$ it is already greater than one. Thus degree six is the only possible exceptional degree for an <group automorphism> not preserving <transpositions>.
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