CyclicCrossings

Pairs (i, j) with i, p(j), p(i), j in cyclic order.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001537via Permutation✓↗

CyclicCrossings(p)Pairs (i, j) with i, p(j), p(i), j in cyclic order.

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 Sum(Map(Function(Count(Filter(
                                                        Range(1, Length(_x)),
                                                        Function(i < _x[j] && _x[j] <
                                                                              _x[i] &&
                                                                 _x[i] < j || _x[j] <
                                                                     _x[i] && _x[i] <
                                                                         j && j <
                                                                             i ||
                                                                 _x[i] < j && j <
                                                                              i &&
                                                                 i < _x[j] || j < i && i <
                                                                              _x[j] &&
                                                                     _x[j] <
                                                                     _x[i], j),
                                                      )), i), Range(
                                                                    1,
                                                                    Length(_x),
                                                                  )))

Examples

Basic #

the one-line word $312$

live ↗

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

the one-line word $2413$

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St001537

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St001537

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St001537

About

Examples	5
Crosswalk	FindStat