  Let $\mathsf{r}_k$ be the unique positive root of $x^k - (x+1)^{k-1} = 0$. We
prove the best known bounds on the number $n_{g,d}$ of $d$-dimensional
generalized numerical semigroups, in particular that
  \[n_{g,d} > C_d^{g^{(d-1)/d}} \mathsf{r}_{2^d}^g\] for some constant $C_d >
0$, which can be made explicit. To do this, we extend the notion of
multiplicity and depth to generalized numerical semigroups and show our lower
bound is sharp for semigroups of depth 2. We also show other bounds on special
classes of semigroups by introducing partition labelings, which extend the
notion of Kunz words to the general setting.
