The h-vector of a simplicial complex

Question

Let $S$ be a polynomial ring over a field. I want to find an ideal $ I\subseteq S$ such that $(1,2,3,1,1,1)$ is the $h$-vector of $S/I$.

We have a relation between $f$-vector and $h$-vector and I found $f$-vector. How can I find $I$? The $f$-vector is $(1,7,21,32,25,9)$. Please help me. Thanks.

Answer 1

Hint: Take $S$ a polynomial ring (over a field) in two variables $x,y$ and $I=(x^3,\dots)$.