A368943 Number of unlabeled mappings from n points to themselves with unique square root (endofunctions).

1, 1, 1, 1, 3, 7, 11, 23, 50

Offset: 0

Author

Keith J. Bauer, Jan 11 2024

Keywords

• nonn
• hard
• more

Comments

A mapping f has a unique square root if there exists a unique g such that gg = f.

Two mappings (endofunctions) are taken to be equivalent up to labeling if one is the conjugation of the other by a permutation. (Conjugation is applying the inverse permutation, the endofunction, and then the permutation, in that order. This is equivalent to permuting the "labels" of the set.)

Examples

			For n = 4, representatives of the a(4) = 3 mappings up to relabeling are
  1->1 2->1 3->2 4->1
  1->2 2->3 3->1 4->1
  1->2 2->3 3->1 4->4
whose unique square roots are respectively
  1->1 2->1 3->4 4->2
  1->3 2->1 3->2 4->2
  1->3 2->1 3->2 4->4
		

Links

• Keith J. Bauer, Visualization of a(6)

Crossrefs

The labeled version is A368867.

Cf. A000700 (permutations only) < this sequence < A368830 (any square maps) < A001372 (all maps).

Extensions

a(8) from Andrew Howroyd, Jan 10 2024