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 An (positive braid monoids) tend to 3.233636 . .. as n tends to infinity. This number is well-known, as it is the growth rate of the coefficients of the only formal power series x0(y) = -(1 + y + 2y2 + 4y3 + 9y4 + center dot center dot center dot ) which is the leading root of the classical partial theta function. We also describe the sequence 1, 1, 2, 4, 9, ... formed by the coefficients of -x0(y), by showing that its kth term (the coefficient of yk) is equal to the number of braids of length k, in the positive braid monoid A infinity on an infinite number of strands, whose maximal lexicographic representative starts