Endofunctions

A name from the enumeratio catalogue with no kernel of its own yet; this page carries its crosswalk only.

Catalogued in the enumeratio database, with crosswalk rows in mathlib4, oeis; not yet written up here.

✓ checked against that system · ! checked, with disagreements (hover a chip)OEISA000312✓↗mathlib4Function.End (Fin n)↗

Endofunctions(...)Catalogued in the enumeratio database, with crosswalk rows in mathlib4, oeis; not yet written up here.

Typed signatures

Type	Package	Applies
(integer<0..>) -> indexed_collection<endofunction>	combinatorics	always

Domain: Combinatorics

Definition

Epsil, as the implementation runs it.

count(_n)

1 if _n == 0 else _n ^ _n

unrank(_n, _r)

Map(
     Function((_r - _r % _n ^ (_n - j)) / _n ^ (_n - j) % _n + 1, j),
     Range(1, _n, 1),
   )

rank(_n, _x)

Fold(Function(acc * _n + (_x[j] - 1), acc, j), 0, Range(1, _n, 1))

valid(_n, _x)

Length(_x) == _n && Fold(Function(
                                   ok_j && 1 <= _x[j] && _x[j] <= _n + 1 - 1,
                                   ok_j,
                                   j,
                                 ), True, Range(1, Length(_x), 1))

Maps

Into and out of Endofunction, called as CombinatorialMap(x, "Name"). Properties are the laws we state and what FindStat records.

To Endofunction

• ArcRepresentationSetPartition → EndofunctionEach position linked to the next in its block, or to itself when last.

Enumeration

Examples

Basic #

$3^3 = 27$ maps from $\{1,2,3\}$ to itself

live ↗

In$\mathrm{Count}(\mathrm{Endofunctions}(3))$Out$27$

The first endofunction of 2, written by its images

live ↗

In$\mathrm{Endofunctions}(2)[1]$Out${\operatorname{Endofunction}}([1, 1])$

Properties #

All four, in enumeration order

live ↗

In$\mathrm{Take}(\mathrm{Endofunctions}(2), 4)$Out$\bigl\lbrack{\operatorname{Endofunction}}([1, 1]), {\operatorname{Endofunction}}([1, 2]), {\operatorname{Endofunction}}([2, 1]), {\operatorname{Endofunction}}([2, 2])\bigr\rbrack$

About

Examples	3
Crosswalk	OEISmathlib4