PositiveBeforeLargerNegativePairCount

Pairs i < j with 0 < p(i) < -p(j): the occurrences of the signed pattern 1-2.

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PositiveBeforeLargerNegativePairCount(p)Pairs i < j with 0 < p(i) < -p(j): the occurrences of the signed pattern 1-2.

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 St001771. A positive entry followed, later, by a negative entry of larger absolute value.

Definition

Epsil, as the implementation runs it.

on SignedPermutation, over _x

0 if Length(_x) < 1 else Max(Sum(Map(
                                      Function(
                                                Count(Filter(
                                                              Range(
                                                                     1,
                                                                     Length(_x),
                                                                   ),
                                                              Function(
                                                                        j > i &&
                                                                        _x[i] >
                                                                        0 && _x[i] <
                                                                            -_x[j],
                                                                        j,
                                                                      ),
                                                            )),
                                                i,
                                              ),
                                      Range(1, Length(_x)),
                                    )), 0)

Examples

Basic #

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

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St001771

FindStat's value for [-1]

live ↗

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

Known value · FindStat St001771

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St001771

About

Examples	4
Crosswalk	FindStat