FixedPoints

The number of fixed points of $p$: positions $i$ with $p_i = i$.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaFixed point (mathematics)↗MathWorldFixedPoint↗WikidataQ217608↗Encyclopedia of Mathematicsfixed_point↗nLabfixed point↗Britannicascience/invariant-point↗FindStatSt001903via ParkingFunction✓↗St000022via Permutation✓↗

FixedPoints(p)the fixed-point count of a one-line permutation $p$

SageMathenumeratio_stat("Permutation", p, "FixedPoints")

Typed signatures

Type	Package	Applies
(permutation) -> integer	combinatorics	always

Domain: Permutation statistics

Details

• A permutation with no fixed points is a derangement (see Derangements)
• Averaged over $S_n$ the count is exactly $1$, independent of $n$
• Defined over Permutation as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
• Takes a Permutation and nothing else — it reads values against their positions, or walks the orbits, so it needs the bijection. Applying it to a bare list is a type error, not a wrong answer.

Definition

Epsil, as the implementation runs it.

on Permutation, over _x

0 if Length(_x) < 1 else Count(Filter(
                                       Range(1, Length(_x)),
                                       Function(_x[i] == i, i),
                                     ))

Examples

Basic #

only $1$ is fixed

live ↗

In$\mathrm{FixedPoints}({\operatorname{Permutation}}([1, 3, 2]))$Out$1$

the identity fixes all

live ↗

In$\mathrm{FixedPoints}({\operatorname{Permutation}}([1, 2, 3]))$Out$3$

the one-line word $312$

live ↗

In$\mathrm{FixedPoints}({\operatorname{Permutation}}([3, 1, 2]))$Out$0$

the one-line word $2413$

live ↗

In$\mathrm{FixedPoints}({\operatorname{Permutation}}([2, 4, 1, 3]))$Out$0$

the endofunction $1 \mapsto 1, 2 \mapsto 3, 3 \mapsto 3$

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{Endofunction}}([1, 3, 3]), \text{FixedPoints})$Out$2$

the signed permutation $1\,\bar{2}\,3$: $\bar{2}$ is not fixed

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{SignedPermutation}}([1, -2, 3]), \text{FixedPoints})$Out$2$

Properties #

a derangement fixes nothing

live ↗

In$\mathrm{FixedPoints}({\operatorname{Permutation}}([2, 3, 1]))$Out$0$

FindStat's value for [1]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{Permutation}}([1]), \text{FixedPoints})$Out$1$

Known value · FindStat St000022

FindStat's value for [1,2]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{Permutation}}([1, 2]), \text{FixedPoints})$Out$2$

Known value · FindStat St000022

FindStat's value for [2,1]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{Permutation}}([2, 1]), \text{FixedPoints})$Out$0$

Known value · FindStat St000022

About

Examples	10
Checked against	SageMath 8 agreemathlib4 5 agree
Crosswalk	WikipediaMathWorldWikidataEncyclopedia of MathematicsnLabBritannicaFindStat ×2SageMath

Implementation

mappedsagepackages/oracle/src/mappings.ts

mappedmathlib4packages/oracle/src/mappings.ts

A 1-based one-line permutation list, rebuilt as an Equiv.Perm (Fin n) (a list that is not a permutation prints "not a permutation"); fixed points counted over Fin n

See also: Derangements, CycleCount, SymmetricGroup