The growth of free inverse monoids

We compute the rate of exponential growth of the free inverse monoid of rank r (and hence an upper bound on the corresponding rate for all r-generated inverse monoids and semigroups). This turns out to be an algebraic number strictly between the obvious bounds of 2r-1 and 2r, tending to 2r as the rank tends to infinity. We also find an explicit expression for the exponential growth rate of the number of idempotents, and prove that this tends to e (2k-1) as k.