On the growth of Artin--Tits monoids and the partial theta function

We present a new procedure to determine the growth function of a homogeneous Garside monoid, with respect to the finite generating set formed by the atoms. In particular, we present a formula for the growth function of each Artin--Tits monoid of spherical type (hence of each braid monoid) with respect to the standard generators, as the inverse of the determinant of a very simple matrix. Using this approach, we show that the exponential growth rates of the Artin--Tits monoids of type $A_n$ (positive braid monoids) tend to $3.233636\ldots$ as $n$ tends to infinity. This number is well-known, as it is the growth rate of the coefficients of the only solution $x_0(y)=-(1+y+2y^2+4y^3+9y^4+\cdots)$ to the classical partial theta function. We also describe the sequence $1,1,2,4,9,\ldots$ formed by the coefficients of $-x_0(y)$, by showing that its $k$th term (the coefficient of $y^k$) is equal to the number of braids of length $k$, in the positive braid monoid $A_{\infty}$ on an infinite number of strands, whose maximal lexicographic representative starts with the first generator $a_1$. This is an unexpected connection between the partial theta function and the theory of braids.