Let $W$ be a Coxeter group (finite or infinite) with (finite) set $S$ of Coxeter generators, and let $I \subseteq S$ be some subset. If $w\in W$ then I call $m_I(w)$ the minimum total number of occurrences of elements of $I$ in some expression for $w$ as a product of elements of $S$, or formally: $$ m_I(w) := \min\{m : \exists x_1,\ldots,x_l\in S\; (w = x_1\cdots x_l \,\land\, m=\#\{i : x_i\in I\})\} $$ Evidently, $m_S$ is the usual length function on $W$, and also, $m_I(w)$ is the minimum total number of occurrences of elements of $I$ in some reduced expression for $w$ (since any expression for $w$ has a reduced subexpression).
I don't know anything else about $m_I$, including:
Question 0: Does this function have a standard name?
Now consider the would-be-Poincaré series associated to $m_I$, namely:
$$ P_I(q) := \sum_{w\in W} q^{m_I(w)} $$
This doesn't make sense in general, but if the ("parabolic") subgroup generated by $S\setminus I$ is finite, which I now assume, then there are only finitely many $w \in W$ having a given value of $m_I$, and $P_I \in \mathbb{Z}[[q]]$.
Question 1: Is this enumerating function rational? (I.e., does it belong to $\mathbb{Q}(q)$?)
Question 2: Assuming yes, how can I compute (a rational form for) it algorithmically?
I'm thinking maybe there's a standard reduced form for elements of $W$ that guarantees that they have the minimum number of elements of $I$, and maybe this standard form is recognizable by a finite automaton, which would be very much in line with other results about Coxeter groups, but I didn't find anything (presumably for lack of knowledge of the correct name for $m_I$). Of course, it would be even better to be able to compute $P_I$ without going through an automaton (along the lines of proposition 7.1.7 in Björner & Brenti's book Combinatorics of Coxeter Groups).
