Counting the number of zero-cycles supported on a set of closed points

Question

I am trying to understand a statement in the paper Random hypersurfaces and embedding curves in surfaces over finite fields by Joseph Gunther.

Let $X$ be a scheme of finite type over $\mathbb{F}_q$ and define $$Z_X (t) = \sum_{n=0}^{\infty} |(\mathrm{Sym}^n X)(\mathbb{F}_q)|t^n.$$ The points of $\mathrm{Sym}^n X$ correspond to formal sums of $n$ points on $X$, with possible repetition; let $\mathrm{Sym}_{[l]}^n X$ be the natural subset comprising just those sums supported on exactly $l$ geometric points. Analogously, define $$Z^{[l]}_X (t) = \sum_{n=0}^{\infty} |(\mathrm{Sym}_{[l]}^n X)(\mathbb{F}_q)|t^n.$$

Fix a value of $l\geq 1$. Suppose we have $r$ distinct closed points $\{P_1 \dots, P_r \}$ of $X$ of any degrees $\lambda_1,\dots,\lambda_r$ such that $\sum \lambda_i = l$. Then the contribution of zero-cycles supported on exactly this set to $Z^{[l]}_X (t)$ is $\prod_{i=1}^{r}(\sum_{n=1}^{\infty}t^{n\lambda_i})$.

I don't understand the last statement. Why is the contribution $\prod_{i=1}^{r}(\sum_{n=1}^{\infty}t^{n\lambda_i})$? Here are some things I do know.

1. By formal sum of $n$ points on $X$, they mean a formal sum $\sum n_p p$ of points over $\overline{\mathbb{F}_p}$ where $n_p \ge 0$ are non-negative integers summing to $n$ and the sum is closed under Galois action. See this answer.

2. There is a bijection between closed points of $X$ and geometric points of $X$ modulo galois action. See this answer. For every closed point of degree $d$, we have $d$ geometric points associated to it.

Answer 1

First, a small correction: the indexing on sum inside the product in the final quoted line should start from $n=0$. (For instance, the empty set in the $0$th symmetric product is a $\Bbb F_q$ point, but isn't counted by the current expression.) Now on to the actual issues.

A formal sum of $n$ geometric points is stable under the Galois action iff it can be written as a formal sum of Galois orbits. Such a formal sum of orbits exactly gives an $\Bbb F_q$ point of the symmetric space, because it's stable under the Galois action. Therefore the counting problem for $\Bbb F_q$-rational zero cycles corresponding to $n$ points supported on our set $\{P_1,\cdots,P_r\}$ is the same as counting ways to write $n$ as a sum of the integers $\lambda_1,\cdots,\lambda_r$ with repetition and disregarding order. But this is exactly what the given expression does: writing $\sum_{a=0}^\infty c_at^a= \prod_{i=1}^{r}(\sum_{n=1}^{\infty}t^{n\lambda_i})$, $c_a$ is exactly the number of ways one can write $a=n_1\lambda_1+\cdots+n_r\lambda_r$ for non-negative values of $n_1,\cdots,n_r$.