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on Tableaux problem 24

The statement

24.[Stan1999a]*Ch. 6 Exer. 25(i) Let the symmetric group $\mathfrak{S}_{n}$ act on the polynomial ring $A=\mathbb{C}[x_{1},\ldots,x_{n},y_{1},\ldots,y_{n}]$ by permuting both the $x_{\bullet}$ and the $y_{\bullet}$ simultaneously; $w\cdot f(x_{1},\ldots,x_{n},y_{1},\ldots,y_{n})=f(x_{w(1)},\ldots,x_{w(n)},y_{w(1)},\ldots,y_{w(n)})$ for all $w\in\mathfrak{S}_{n}$ . Let $I$ be the ideal generated by all invariants of positive degree, i.e.,

I=\left<f\in A:w\cdot f=f\textup{ for all }w\in\mathfrak{S}_{n}\textup{, and }f(0)=0\right>.

Prove that the Catalan number $C_{n}=\frac{1}{n+1}\binom{2n}{n}$ is the dimension of the subspace of $A/I$ affording the sign representation, i.e.,

C_{n}=\operatorname{dim}\{f\in A/I:w\cdot f=(\operatorname{sgn}w)f\textup{ for all }w\in\mathfrak{S}_{n}\}.

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.