FixedPointsBeforeLargestMovedPoint

Fixed points smaller than the largest i with p(i) != i.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001810via Permutation✓↗

FixedPointsBeforeLargestMovedPoint(p)Fixed points smaller than the largest i with p(i) != i.

Typed signatures

Type	Package	Applies
(permutation) -> number	combinatorics	always

Domain: Permutation statistics

Details

• 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,
                                             Max(Join(
                                                       [0],
                                                       Filter(
                                                               Range(
                                                                      1,
                                                                      Length(_x),
                                                                    ),
                                                               Function(_x[i] <
                                                                        i || _x[i] >
                                                                            i, i),
                                                             ),
                                                     )),
                                             1,
                                           ), Function(_x[i] == i, i)))

Examples

Basic #

the one-line word $312$

live ↗

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

the one-line word $2413$

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St001810

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St001810

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St001810

About

Examples	5
Crosswalk	FindStat