Fertility numbers

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 $3 \operatorname{modulo} 4$. 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.

Keywords

permutation, stack-sorting, fertility, valid hook configuration

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The author was supported by a Fannie and John Hertz Foundation Fellowship and an NSF Graduate Research Fellowship.

Received 29 April 2019

Accepted 10 July 2019

Published 11 May 2020