Consider the ring $k[x\_1,x\_2,x\_3,x\_4]/(x\_1x\_3,x\_1x\_4,x\_2x\_3,x\_2x\_4)$. How do I find the dimension of the vector space of homogeneous representatives of the quotient ring of each degree?

I've only managed to exhibit bases for low degrees:

0: $1$; dimension 1

1: $x\_1,x\_2,x\_3,x\_4$; dimension 4

2: $x\_1^2,x\_2^2,x\_3^2,x\_4^2,x\_1x\_2,x\_3x\_4$; dimension 6

3: $x\_1^3,x\_2^3,x\_3^3,x\_4^3, x\_1^2x\_2,x\_1x\_2^2, x\_3^2x\_4,x\_3x\_4^2$; dimension 8

So my conjecture is that the dimension of polynomials of degree $k>0$ is $2k+2$. Is it true? How to prove it?