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… 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 $x_0(y)=-(1+y+2y^2+4y^3+9y^4+\cdots )$ which is the leading root of the classical partial theta function.

We also describe the sequence $1,1,2,4,9,\dots$ formed by the coefficients of $-x_0(y)$, by showing that its kth 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.

我们提出了一个新的程序来确定一个齐次 Garside 幺半群的增长函数，关于由原子形成的有限生成集。特别是，我们提出了每个球形类型的 Artin-Tits 幺半群（因此每个辫子幺半群）相对于标准生成器的增长函数的公式，作为非常简单矩阵的行列式的逆。

使用这种方法，我们证明了 Artin-Tits 类群的指数增长率${\mathrm{一种}}_n$（正辫子幺半群）趋向于 3.233636... 因为n趋于无穷大。这个数字是众所周知的，因为它是唯一正式幂级数的系数的增长率$X_0(\mathrm{是的})=-(1+\mathrm{是的}+2{\mathrm{是的}}^2+4{\mathrm{是的}}^3+9{\mathrm{是的}}^4+\cdots )$它是经典偏 theta 函数的主根。

我们还描述了序列$1,1,2,4,9,\dots$由的系数形成$-X_0(\mathrm{是的})$，通过证明它的第k项（系数${\mathrm{是的}}^{ķ}$) 等于长度为k的辫子的数量，在正辫子幺半群中${\mathrm{一种}}_{\infty }$在无限数量的链上，其最大词典代表从第一个生成器开始${\mathrm{一种}}_1$. 这是部分 theta 函数和辫子理论之间出乎意料的联系。