Hilbert function of a primary ideal

I'm reading Geramita's lectures on fat points, but I found a passage a bit confusing. (I'll post the reference as soon as I'll find the link). I read that a fat point (of order $t$, defined by the ${p}$-primary ideal $p^t$) in $\mathbb{P}^n$ behaves like $\binom{n+t-1}{n}$ distinct points (in the sense of the Hilbert function).

In order to prove this, the paper states that $$H(S/p^t,s) = \binom{n+t-1}{n}$$ if $s\geq t$, with $S=k[x_0,x_1,\ldots,x_n]$.

To prove this observation, they observe that $F \in p^{t+1}$ $\iff$ all the partial derivatives of order $\leq t$, vanish at $P$ (the point defined by the ideal).

But I can't understand why $H(S/p^t)$ is equal to that binomial coefficient, I mean it should be equal to $$H(R/p^t,s)=\dim_k R_s-\dim_k (p^t)_s=\binom{n+s}{s}-\dim_k (p^t)_s$$ and I can't find an easy formula for the second dimension, I guess it involves some combinatoric tool I can't see. Any help would be much appreciate.

• If someone is interested this is the link ti the paper google.com/url?q=https://www3.nd.edu/~stolz/Math70330(F2008)/… – konoa May 14 '18 at 9:26
• Where is the claim/theorem in the note? – Youngsu May 18 '18 at 0:14
• at the bottom of page 4, but in the last days I figured out why it works. Thanks anyway. – konoa May 22 '18 at 10:25
• Great! Thanks for the reply. – Youngsu May 22 '18 at 10:40