DescentsAfterZeroingFixedPoints

The descents of the word with every fixed point replaced by 0.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000333via Permutation✓↗

DescentsAfterZeroingFixedPoints(p)The descents of the word with every fixed point replaced by 0.

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) < 2 else Apply(
                                (w: list<integer>) => Sum(Map(
                                                               Function(1, i),
                                                               Filter(
                                                                       Range(
                                                                              1,
                                                                              Length(w) -
                                                                              1,
                                                                              1,
                                                                            ),
                                                                       Function(w[i] >
                                                                                w[i + 1], i),
                                                                     ),
                                                             )),
                                Map(
                                     Function(0 if _x[i] == i else _x[i], i),
                                     Range(1, Length(_x)),
                                   ),
                              )

Examples

Basic #

the one-line word $312$

live ↗

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

the one-line word $2413$

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St000333

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St000333

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St000333

About

Examples	5
Crosswalk	FindStat