Show that $\sum_{n=0}^\infty (order\ {S_n})q^n=\prod_{m\ge 1}(1-q^m)^{-1}$ Let $T=\mathbb (C^*)^2$ acts on $\mathbb C[x,y]$ via $(t_1,t_2)(x,y)=(t_1x,t_2y)$, let $S_n$ be the set of ideals $I$ of $\mathbb C[x,y]$ such that $TI=I$ and $\mathbb C[x,y]/I$ is $n$-dimensional $\mathbb C$-vector space. If $order\ S_{0}=0$. show that $$\sum_{n=0}^\infty (order\ {S_n})q^n=\prod_{m\ge 1}(1-q^m)^{-1}.$$

Since this looks like a homework problem let me just give an outline.


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*Prove that an ideal is invariant under the torus action if and only if it is generated by monomials.

*Prove that monomial ideals $I \subset \mathbb C[x,y]$ can be identified with partitions. Hint: draw a square grid with squares $(i,j)_{i \geq 0, j \geq 0}$. Mark each square $(i,j)$ such that $x^i y^j \in I$ with a different color. What kind of shape do you get?

*Apply Euler's generating function for partitions.

