PositiveBeforeNegativeCount

Twice the adjacent pairs of a positive entry followed by a negative one, plus 1 when the first entry is negative.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001870via SignedPermutation✓↗

PositiveBeforeNegativeCount(p)Twice the adjacent pairs of a positive entry followed by a negative one, plus 1 when the first entry is negative.

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 St001870 counts them in the word p(-n), ..., p(-1), p(1), ..., p(n) with p(-i) = -p(i): each adjacent pair twice, once reflected, and the pair across the middle when p(1) is negative.

Definition

Epsil, as the implementation runs it.

on SignedPermutation, over _x

0 if Length(_x) < 1 else 2 * (0 if Length(_x) < 2 else Count(Filter(
                                                                     Range(
                                                                            1,
                                                                            Length(_x) -
                                                                            1,
                                                                          ),
                                                                     Function(_x[i] >
                                                                              0 &&
                                                                              _x[i + 1] <
                                                                              0, i),
                                                                   ))) +
                         (1 if _x[1] < 0 else 0)

Examples

Basic #

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

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St001870

FindStat's value for [-1]

live ↗

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

Known value · FindStat St001870

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St001870

About

Examples	4
Crosswalk	FindStat