Let $k[x]=k[x_1,\dots,x_n]$ be the set of polynomials in $n$ variables. Given a $k
I guessed it would be $\frac{k(k+1)\cdots(k+n-1)}{n!}$, with the $n!$ since you have various choices of bases. I am having trouble justifying this intuition, or even confirming if it is correct.