TITLE: Which ideals have standard Hilbert series?
QUESTION [8 upvotes]: Let $m$ and $d$ be two positive integers. 
Consider the polynomial ring $R = \mathbb{C}[x_1 , \dots , x_m]$. Let $I$ be an ideal of $R$ generated by a finite family of polynomials of degree $d$, and finally let $H(t)$ be the Hilbert series of $R/I$. 
I say that the Hilbert series $H(t)$ is standard if there exists an integer $n$ such that $$H(t) = \frac{(1-t^d)^n}{(1-t)^m} \, . $$
My question
I am looking for a (ideally, necessary and sufficient) condition on $I$ for its Hilbert series to be standard. 
An example
Consider the case $m=3$ and $d=2$. Then 

For $I=\langle x_1 x_2 \rangle$, we have $H(t) = \frac{1-t^2}{(1-t)^3}$, so $I$ satisfies the condition. 
For $I=\langle x_1 x_2 , x_1 x_3 \rangle$ we have $H(t) = \frac{t^3-2 t^2+1}{(1-t)^3}$ and this can not be put in standard form, so $I$ does not satisfy the condition.

REPLY [2 votes]: I assume you mean that I is generated by homogeneous polynomials of degree d. If so, a necessary and sufficient condition is that I be generated by a regular sequence of n polynomials of degree d. See Corollary 3.3 in Stanley (Advances in Math. 28 (1978) 57-83).