A368830 Number of square unlabeled endofunctions from n points to themselves.

1, 1, 2, 4, 9, 20, 47, 105

Offset: 0

Author

Keith J. Bauer, Jan 07 2024

Keywords

• nonn
• hard
• more

Comments

Also known as square maps or square mapping patterns.

Two 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

			The a(3) = 4 square endofunctions are:
  1->1, 2->2, 3->3
  1->1, 2->1, 3->1 (equivalent to any constant function)
  1->1, 2->2, 3->1 (equivalent to any function consisting of 2 1-cycles)
  1->2, 2->3, 3->1 (equivalent to any 3-cycle)
Each function listed here is its own square root, except for the 3-cycle, whose square root is its inverse.
		

Crossrefs

Cf. A001372, A102687 (labeled version).