  A nonnegative integer is called a fertility number if it is equal to the
number of preimages of a permutation under West's stack-sorting map. We prove
structural results concerning permutations, allowing us to deduce information
about the set of fertility numbers. In particular, the set of fertility numbers
is closed under multiplication and contains every nonnegative integer that is
not congruent to $3$ modulo $4$. We show that the lower asymptotic density of
the set of fertility numbers is at least $1954/2565\approx 0.7618$. We also
exhibit some positive integers that are not fertility numbers and conjecture
that there are infinitely many such numbers.
