OFFSET
1,3
COMMENTS
The fertility of a permutation pi is |s^{-1}(pi)|, where s is West's stack-sorting map. A nonnegative integer is called a fertility number if it is the fertility of some permutation.
The set of fertility numbers is closed under multiplication.
Every nonnegative integer that is not congruent to 3 modulo 4 is a fertility number.
The lower asymptotic density of this sequence is at least 0.7618. In particular, there are infinitely many fertility numbers that are congruent to 3 modulo 4. The smallest of these is 27. It appears as though 95 is the second-smallest fertility number that is congruent to 3 modulo 4.
It is conjectured that there are infinitely many positive integers that are not fertility numbers.
Empirically found 149 terms congruent 3 mod 4, the second smallest being 39 followed by 51, 63, 95, 123, ... - Jon Maiga, Oct 28 2018
LINKS
C. Defant, Fertility numbers, arXiv:1809:04421 [math.CO], 2018.
J. Maiga, Stack sorting and fertility numbers, 2018.
EXAMPLE
The preimages of 123 under the stack-sorting map are 123, 132, 213, 312, and 321. This shows that the fertility of 123 is 5, so 5 is a fertility number.
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Colin Defant, Sep 14 2018
STATUS
approved