CycleMinimumExceedancesDeficiencies

The sum over the cycles of the lesser of the number of exceedances and the number of the other points.

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CycleMinimumExceedancesDeficiencies(p)The sum over the cycles of the lesser of the number of exceedances and the number of the other points.

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 Max(Sum(Map(
                                      Function(Min([
                                                     Count(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 &&
                                                                            _x[m] >
                                                                            m, m),
                                                                 )),
                                                     Count(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 &&
                                                                            _x[m] <=
                                                                            m, 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),
                                            ),
                                    )), 0)

Examples

Basic #

the one-line word $312$

live ↗

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

the one-line word $2413$

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St001269

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St001269

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St001269

About

Examples	5
Crosswalk	FindStat