AdjacentCycles

Cycles of the form (a, a+1, ..., a+k), fixed points included.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000153via Permutation✓↗

AdjacentCycles(p)Cycles of the form (a, a+1, ..., a+k), fixed points included.

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(Count(Filter(
                                                              Range(
                                                                     i,
                                                                     Length(_x),
                                                                     1,
                                                                   ),
                                                              Function(_x[m] ==
                                                                       i && Count(Filter(
                                                                                         Range(i, Length(_x), 1),
                                                                                         Function(j < m &&
                                                                                                  (_x[j] <
                                                                                                   j + 1 ||
                                                                                                   _x[j] >
                                                                                                   j + 1), j),
                                                                                       )) ==
                                                                           0, m),
                                                            )) > 0, i),
                                     ))

Examples

Basic #

the one-line word $312$

live ↗

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

the one-line word $2413$

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St000153

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St000153

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St000153

About

Examples	5
Crosswalk	FindStat