FlagDenertIndex

The negative entries plus twice the Denert index of the permutation |p(1)| ... |p(n)|.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001819via SignedPermutation✓↗

FlagDenertIndex(p)The negative entries plus twice the Denert index of the permutation |p(1)| ... |p(n)|.

Typed signatures

Type	Package	Applies
(signed_permutation) -> number	combinatorics	always

Domain: Signed permutation statistics

Details

• Defined over SignedPermutation as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
• Takes a SignedPermutation 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.
• FindStat's St001819.

Definition

Epsil, as the implementation runs it.

on SignedPermutation, over _x

0 if Length(_x) < 1 else Count(Filter(
                                       Range(1, Length(_x)),
                                       Function(_x[i] < 0, i),
                                     )) + 2 * Apply(
                                                    (w: list<integer>) => Sum(Filter(
                                                                                      Range(1, Length(w)),
                                                                                      Function(w[i] > i, i),
                                                                                    )) +
                                                                          Sum(Map(
                                                                                   Function(Count(Filter(
                                                                                                          Range(i + 1, Length(w)),
                                                                                                          Function(w[j] > j &&
                                                                                                                   w[i] > w[j], j),
                                                                                                        )) if w[i] > i else 0, i),
                                                                                   Range(1, Length(w)),
                                                                                 )) +
                                                                          Sum(Map(
                                                                                   Function(Count(Filter(
                                                                                                          Range(i + 1, Length(w)),
                                                                                                          Function(w[j] <= j &&
                                                                                                                   w[i] > w[j], j),
                                                                                                        )) if w[i] <= i else 0, i),
                                                                                   Range(1, Length(w)),
                                                                                 )),
                                                    Map(
                                                         Function(Abs(_x[i]), i),
                                                         Range(1, Length(_x)),
                                                       ),
                                                  )

Examples

Basic #

the signed word $2\,\bar{1}\,\bar{3}$

live ↗

In$\mathrm{FlagDenertIndex}({\operatorname{SignedPermutation}}([2, -1, -3]))$Out$4$

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St001819

FindStat's value for [-1]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{SignedPermutation}}([-1]), \text{FlagDenertIndex})$Out$1$

Known value · FindStat St001819

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St001819

About

Examples	4
Crosswalk	FindStat