h-vector of an unmixed ideal We know that:


*

*if a ring is Cohen-Macaulay, then it is unmixed. But the converse is not true.

*if a ring is Cohen-Macaulay, then its $h$-vector is positive. The converse is not true. 


Then I expect to find unmixed ideals that have positive $h$-vector, but they are not CM.  Now the question is: 

Is it possible to find an example of a not unmixed ideal which has positive $h$-vector?


I solved my question starting from the simplicial complex $$\Delta = \{\{x_1,x_2\},\{x_1,x_3\},\{x_1,x_4\},\{x_2,x_3\},\{x_2,x_4\},x_5\}$$
It is not unmixed but $h=(1,3,1)$, since $f=(1,5,5)$. 
So, the ideal that I was looking for is
$$I_{\Delta} = (x_1x_2x_3, x_1x_2x_4,x_3x_4, x_1x_5,x_2x_5,x_3x_5, x_4x_5) \subseteq \mathbb{K}[x_1, \dots, x_5]$$
