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

0,5

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.)

Links

Table of n, a(n) for n=0..8.

Keith J. Bauer, Visualization of a(6)

Example

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

Crossrefs

The labeled version is A368867.

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

Sequence in context: A116606 A188132 A139814 * A099902 A336897 A316962

Adjacent sequences: A368940 A368941 A368942 * A368944 A368945 A368946

Keyword

nonn,hard,more

Author

Keith J. Bauer, Jan 11 2024

Extensions

a(8) from Andrew Howroyd, Jan 10 2024

Status

approved