A317551 Fertility numbers.

0, 1, 2, 4, 5, 6, 8, 9, 10, 12, 13, 14, 16, 17, 18, 20, 21, 22, 24, 25, 26, 27, 28, 29, 30

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

Table of n, a(n) for n=1..25.

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

Sequence in context: A285601 A139255 A277676 * A004773 A104401 A184421

Adjacent sequences: A317548 A317549 A317550 * A317552 A317553 A317554

Keyword

nonn,more

Author

Colin Defant, Sep 14 2018

Status

approved