OneBoxPatterns

Positions i with a neighbouring entry, left or right, differing from p(i) by exactly 1.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000064via Permutation✓↗

OneBoxPatterns(p)Positions i with a neighbouring entry, left or right, differing from p(i) by exactly 1.

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 Count(Filter(Range(1, Length(_x)), Function(
                                                                      (Abs(_x[i] -
                                                                           _x[i -
                                                                              1]) ==
                                                                       1 if i > 1 else False) ||
                                                                      (Abs(_x[i] -
                                                                           _x[i +
                                                                              1]) ==
                                                                       1 if i <
                                                                           Length(_x) else False),
                                                                      i,
                                                                    )))

Examples

Basic #

the one-line word $312$

live ↗

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

the one-line word $2413$

live ↗

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

Properties #

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St000064

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St000064

FindStat's value for [1,2,3]

live ↗

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

Known value · FindStat St000064

About

Examples	5
Crosswalk	FindStat