I'm having troubles understanding exterior algebra graded structure.
I need to write the polynomial (a sort of Poincare polynomial) given by $P(t)=\sum_{p=0}^{N} (\dim \Lambda^p )\ t^p,$ where $\Lambda$ is an exterior algebra $\Lambda=\Lambda(x_1,...,x_l)$ with generators $x_i$ of degree $d_i,$ and $N=\sum_{i=1}^l d_i.$
Is it correct to say that
$$\Lambda^p \text{ has basis given by } \left\{ x_{i_1} \wedge ... \wedge x_{i_k} \ \vert \ 1 \leq i_1 < ... <i_k \leq l, \ \sum_{m=1}^k d_{i_m}=p \right\}$$
If that's the case, the polynomial $P(t)$ would have the form $$P(t)=\prod_{i=1}^l (1 + t^{d_i})$$ Am I correct?
Thanks to everyone.