Area

The whole squares between the path and the diagonal below it: the sum of the area sequence.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000012via DyckPath✓↗St001295via DyckPath✓↗

Area(path)The whole squares between the path and the diagonal below it: the sum of the area sequence.

Typed signatures

Type	Package	Applies
(dyck_path) -> number	combinatorics	always

Domain: Dyck path statistics

Details

• Defined over DyckPath as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
• Takes a DyckPath 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.
• Haglund's area (FindStat St000012). Half of: the total of the heights after each step, less the semilength.

Definition

Epsil, as the implementation runs it.

on DyckPath, over _x

0 if Length(_x) < 1 else (Sum(Map(Function(Sum(Map(
                                                    Function(2 * _x[k] - 1, k),
                                                    Range(1, i),
                                                  )), i), Range(1, Length(_x)))) -
                          Length(_x) / 2) / 2

Examples

Basic #

the step word $UUDDUD$

live ↗

In$\mathrm{Area}({\operatorname{DyckPath}}([1, 1, 0, 0, 1, 0]))$Out$1$

the step word $UDUUDD$

live ↗

In$\mathrm{Area}({\operatorname{DyckPath}}([1, 0, 1, 1, 0, 0]))$Out$1$

Properties #

FindStat's value for [1,0]

live ↗

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

Known value · FindStat St000012

FindStat's value for [1,0,1,0]

live ↗

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

Known value · FindStat St000012

FindStat's value for [1,1,0,0]

live ↗

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

Known value · FindStat St000012

About

Examples	5
Crosswalk	FindStat ×2