MinimalCycleDifference

The smallest, over the cycles, of the largest difference between two elements of a cycle.

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MinimalCycleDifference(p)The smallest, over the cycles, of the largest difference between two elements of a cycle.

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 Min(Map(
                                  Function(Max(Map(
                                                    Function(Abs(m - i), m),
                                                    Filter(
                                                            Range(1, Length(_x)),
                                                            Function(Count(Filter(
                                                                                   Range(1, Length(_x), 1),
                                                                                   Function(Fold(
                                                                                                  Function(_x[w], w, t),
                                                                                                  i,
                                                                                                  Range(1, k, 1),
                                                                                                ) ==
                                                                                            m, k),
                                                                                 )) >
                                                                     0, m),
                                                          ),
                                                  )), i),
                                  Filter(
                                          Range(1, Length(_x), 1),
                                          Function(Fold(
                                                         Function(
                                                                   w if w <= 0 else 0 if w == i else -1 if w < i else _x[w],
                                                                   w,
                                                                   k,
                                                                 ),
                                                         _x[i],
                                                         Range(1, Length(_x), 1),
                                                       ) == 0, i),
                                        ),
                                ))

Examples

Basic #

the one-line word $312$

live ↗

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

the one-line word $2413$

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St000210

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St000210

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St000210

About

Examples	5
Crosswalk	FindStat