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

Abstract

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+⋯) 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∞ on an infinite number of strands, whose maximal lexicographic representative starts with the first generator a1. This is an unexpected connection between the partial theta function and the theory of braids.