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

We present a new procedure to determine the growth function of homogeneous Garside monoid, with respect finite generating set formed by atoms. In particular, we formula for each Artin–Tits monoid spherical type (hence braid monoid) standard generators, as inverse determinant very simple matrix. Using this approach, show that exponential rates monoids An (positive monoids) tend 3.233636… n tends infinity. This number is well-known, it rate coefficients only formal power series x0(y)=−(1+y+2y2+4y3+9y4+⋯) which leading root classical partial theta function. also describe sequence 1,1,2,4,9,… −x0(y), showing its kth term (the coefficient yk) equal braids length k, in positive A∞ on an infinite strands, whose maximal lexicographic representative starts first generator a1. unexpected connection between and theory braids.