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

Author

Colin Defant, Sep 14 2018

Keywords

• nonn
• more

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

Examples

			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.
		

Links

• C. Defant, Fertility numbers, arXiv:1809:04421 [math.CO], 2018.
• J. Maiga, Stack sorting and fertility numbers, 2018.