FlagWeakExcedance

Twice the weak excedances plus the negative entries.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001817via SignedPermutation✓↗

FlagWeakExcedance(p)Twice the weak excedances plus the negative entries.

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 St001817. A weak excedance is an entry p(i) >= i.

Definition

Epsil, as the implementation runs it.

on SignedPermutation, over _x

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

Examples

Basic #

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

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St001817

FindStat's value for [-1]

live ↗

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

Known value · FindStat St001817

FindStat's value for [1,2]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{SignedPermutation}}([1, 2]), \text{FlagWeakExcedance})$Out$4$

Known value · FindStat St001817

About

Examples	4
Crosswalk	FindStat