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In a 1986 paper, Harer and Zagier proved the recursion: $$(n+1)e(g,n)=(4n-2)e(g,n-1)+(2n-1)(n-1)(2n-3)e(g-1,n-2)$$ where e(g,n) is the number of ways of grouping sides $S_1...S_{2n}$ of a 2n-gon ...

Let $F_1, F_2$ and $F_3$ be finites sets of irreducible polynomials in $\mathbb{C}[x_0, \ldots, x_n]$ of degree at most $2$ such that $F_1 \cap F_2 \cap F_3 = \varnothing$ and for every $Q_1, Q_2$ ...