Closed formulas for two Poincaré series

Question

Associated with an arbitrary direct sum $E = \bigoplus_{i \ge 0} E_i$, of finite dimensional $k$-vector spaces $E_i$, $i = 0, 1, 2, \dots,$ there is a formal power series $P_E$, with nonnegative integer coefficients, called Poincare series:$$P_E(t) = \sum_{i = 0}^\infty \dim E_i \cdot t^i.$$For each $i = 0, 1, 2, \dots,$ let $k^i[x_1, \dots, x_n] \subset k[x_1, \dots, x_n]$, resp. $k^i\langle x_i, \dots, x_n\rangle \subset k\langle x_1, \dots, x_n\rangle$, denote the $k$-linear span of monomials, resp. noncommutative monomials, of total degree $i$ (each of the variables $x_1, \dots, x_n$ is assigned degree $1$). Thus, one has a countable direct sum decomposition (as $k$-vector spaces)$$k[x_1, \dots, x_n] = \bigoplus_{i \ge 0} k^i[x_1, \dots, x_n],\text{ }k\langle x_1, \dots, x_n\rangle = \bigoplus_{i \ge 0} k^i \langle x_1, \dots, x_n\rangle.$$My question is, what are the closed formulas for Poincaré series $P_{\mathbb{C}[x_1, \dots, x_n]}$ and $P_{\mathbb{C}\langle x_1, \dots, x_n\rangle}$?

Answer 1

For the commutative case, we know that

$$ \dim k^i [x_1, \cdots, x_n] = \sum_{e_1 + \cdots + e_n = i} 1. $$

This gives

\begin{align*} P_{\Bbb{C}[x_1, \cdots, x_n]}(t) &= \sum_{i=0}^{\infty} \sum_{e_1 + \cdots + e_n = i} t^i \\ &= \sum_{i=0}^{\infty} \sum_{e_1 + \cdots + e_n = i} t^{e_1} \cdots t^{e_n} \\ &= \sum_{e_1, \cdots, e_n} t^{e_1} \cdots t^{e_n} \\ &= \frac{1}{(1-t)^n}. \end{align*}

The non-commutative case is much easier, so try by yourself!