# Fertility Numbers

• Published in 2018
In the collections
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.

### BibTeX entry

@article{FertilityNumbers,
title = {Fertility Numbers},
abstract = {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.},
url = {http://arxiv.org/abs/1809.04421v3 http://arxiv.org/pdf/1809.04421v3},
year = 2018,
author = {Colin Defant},
comment = {},
urldate = {2020-10-16},
archivePrefix = {arXiv},
eprint = {1809.04421},
primaryClass = {math.CO},
collections = {attention-grabbing-titles,combinatorics,integerology}
}