CyclicMisalignments

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

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001536via Permutation✓↗

CyclicMisalignments(p)Pairs (i, j) with i, p(i), j, p(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[i] &&
                                                                   _x[i] < j && j < _x[j] ||
                                                                   _x[i] < j &&
                                                                   j < _x[j] && _x[j] < i || j < _x[j] &&
                                                                       _x[j] < i && i <
                                                                           _x[i] || _x[j] <
                                                                               i &&
                                                                               i <
                                                                               _x[i] &&
                                                                               _x[i] <
                                                                               j,
                                                                   j,
                                                                 ),
                                                       )),
                                           i,
                                         ), Range(1, Length(_x))))

Examples

Basic #

the one-line word $312$

live ↗

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

the one-line word $2413$

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St001536

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St001536

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St001536

About

Examples	5
Crosswalk	FindStat