CycleDescents

Steps down within the cycles, each written from its smallest point.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000317via Permutation✓↗

CycleDescents(p)Steps down within the cycles, each written from its smallest point.

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, Length(_x)),
                                       Function(_x[i] < i && Fold(
                                                                   Function(
                                                                             w if w <= 0 else 0 if w == _x[i] else -1 if w < _x[i] else _x[w],
                                                                             w,
                                                                             k,
                                                                           ),
                                                                   _x[_x[i]],
                                                                   Range(
                                                                          1,
                                                                          Length(_x),
                                                                          1,
                                                                        ),
                                                                 ) < 0, i),
                                     ))

Examples

Basic #

the one-line word $312$

live ↗

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

the one-line word $2413$

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St000317

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St000317

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St000317

About

Examples	5
Crosswalk	FindStat