Fertility Numbers

TL;DR

This paper investigates fertility numbers, which count preimages of permutations under a specific stack-sorting map, revealing their structural properties, closure under multiplication, and density, along with conjectures about non-fertility numbers.

Contribution

The paper establishes the closure of fertility numbers under multiplication and characterizes their set, including density estimates and the existence of non-fertility numbers.

Findings

Fertility numbers are closed under multiplication.

All nonnegative integers not congruent to 3 mod 4 are fertility numbers.

The lower asymptotic density of fertility numbers is at least approximately 0.7618.

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.