PermutationSupport

The points $p$ moves, in increasing order.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000673Wolfram LanguagePermutationLength!PermutationMax!PermutationMin!PermutationSupport!

PermutationSupport(p)The points

p

moves, in increasing order.

Typed signatures

Type	Package	Applies
`(permutation) -> list<integer>`	`combinatorics`	always

Domain: Permutation statistics

Details

Defined over Permutation as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
Wolfram's Cycles input is not read here: wrap a one-line word in Permutation, or turn Cycles into one with PermutationList first
The identity has an empty support. Its size, largest and smallest points are Length, Max and Min of it: Wolfram's PermutationLength, PermutationMax and PermutationMin are not heads here
Max and Min of an empty list are NaN in compute-engine, not Wolfram's −∞ and ∞. Wolfram's PermutationMax and PermutationMin of the identity are 0 and ∞, so those compositions carry the bound: Max(0, …) and Min(Infinity, …)
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

[] if Length(_x) < 1 else Fold(Function(
                                         Join(a, [b]) if Abs(_x[b] - b) > 0 else a,
                                         a,
                                         b,
                                       ), [], Range(1, Length(_x)))

Examples

Basic #

$(1\,5)(2\,3\,4)$ moves all five points

In

\mathrm{PermutationSupport}({\operatorname{Permutation}}([5, 3, 4, 2, 1]))

Out

\bigl\lbrack1, 2, 3, 4, 5\bigr\rbrack

Known value · wolframscript 15.0

the one-line word $312$

In

\mathrm{PermutationSupport}({\operatorname{Permutation}}([3, 1, 2]))

Out

\bigl\lbrack1, 2, 3\bigr\rbrack

the one-line word $2413$

In

\mathrm{PermutationSupport}({\operatorname{Permutation}}([2, 4, 1, 3]))

Out

\bigl\lbrack1, 2, 3, 4\bigr\rbrack

all three points move

In

\mathrm{Length}(\mathrm{PermutationSupport}({\operatorname{Permutation}}([2, 3, 1])))

Out

3

Known value · wolframscript 15.0

the one-line word $312$

In

\mathrm{Length}(\mathrm{PermutationSupport}({\operatorname{Permutation}}([3, 1, 2])))

Out

3

the one-line word $2413$

In

\mathrm{Length}(\mathrm{PermutationSupport}({\operatorname{Permutation}}([2, 4, 1, 3])))

Out

4

3 and 4 are fixed, so the largest moved point is 2

In

\max(\mathrm{PermutationSupport}({\operatorname{Permutation}}([2, 1, 3, 4])))

Out

2

Known value · wolframscript 15.0

the one-line word $312$

In

\max(\mathrm{PermutationSupport}({\operatorname{Permutation}}([3, 1, 2])))

Out

3

the one-line word $2413$

In

\max(\mathrm{PermutationSupport}({\operatorname{Permutation}}([2, 4, 1, 3])))

Out

4

1 is fixed, so the smallest moved point is 2

In

\min(\mathrm{PermutationSupport}({\operatorname{Permutation}}([1, 3, 2, 4])))

Out

2

Known value · wolframscript 15.0

the one-line word $312$

In

\min(\mathrm{PermutationSupport}({\operatorname{Permutation}}([3, 1, 2])))

Out

1

the one-line word $2413$

In

\min(\mathrm{PermutationSupport}({\operatorname{Permutation}}([2, 4, 1, 3])))

Out

1

Scope #

1 and 4 are fixed

In

\mathrm{PermutationSupport}({\operatorname{Permutation}}([1, 3, 2, 4]))

Out

\bigl\lbrack2, 3\bigr\rbrack

Known value · wolframscript 15.0

only 2 and 3 move

In

\mathrm{Length}(\mathrm{PermutationSupport}({\operatorname{Permutation}}([1, 3, 2, 4])))

Out

2

Known value · wolframscript 15.0

the cycles $(1\,5)(2\,3\,4)$ move five points

In

\mathrm{Length}(\mathrm{PermutationSupport}({\operatorname{Permutation}}([5, 3, 4, 2, 1])))

Out

5

Known value · wolframscript 15.0

$(1\,5)(2\,3\,4)$

In

\max(\mathrm{PermutationSupport}({\operatorname{Permutation}}([5, 3, 4, 2, 1])))

Out

5

Known value · wolframscript 15.0

$(1\,5)(2\,3\,4)$

In

\min(\mathrm{PermutationSupport}({\operatorname{Permutation}}([5, 3, 4, 2, 1])))

Out

1

Known value · wolframscript 15.0

Properties #

the identity moves nothing

In

\mathrm{PermutationSupport}({\operatorname{Permutation}}([1, 2, 3]))

Out

\bigl\lbrack\bigr\rbrack

Known value · wolframscript 15.0

the identity moves nothing

In

\mathrm{Length}(\mathrm{PermutationSupport}({\operatorname{Permutation}}([1, 2, 3])))

Out

0

Known value · wolframscript 15.0

FindStat's value for [1,2]

In

\mathrm{Length}(\mathrm{PermutationSupport}({\operatorname{Permutation}}([1, 2])))

Out

0

Known value · FindStat St000673

FindStat's value for [2,1]

In

\mathrm{Length}(\mathrm{PermutationSupport}({\operatorname{Permutation}}([2, 1])))

Out

2

Known value · FindStat St000673

FindStat's value for [1,2,3]

In

\mathrm{Length}(\mathrm{PermutationSupport}({\operatorname{Permutation}}([1, 2, 3])))

Out

0

Known value · FindStat St000673

the identity moves nothing; the bound 0 is Wolfram's answer

In

\max(0, \mathrm{PermutationSupport}({\operatorname{Permutation}}([1, 2, 3])))

Out

0

Known value · wolframscript 15.0

without the bound, the largest of no points

In

\max(\mathrm{PermutationSupport}({\operatorname{Permutation}}([1, 2, 3])))

Out

\operatorname{NaN}

Known value · wolframscript 15.0

the identity moves nothing, so there is no smallest point; the bound infinity is Wolfram's answer

In

\min(∞, \mathrm{PermutationSupport}({\operatorname{Permutation}}([1, 2, 3])))

Out

∞

Known value · wolframscript 15.0

without the bound, the smallest of no points

In

\min(\mathrm{PermutationSupport}({\operatorname{Permutation}}([1, 2, 3])))

Out

\operatorname{NaN}

Known value · wolframscript 15.0

About

Examples	26
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