LargestAreaSequenceEntry

The largest entry of the area sequence: the greatest area to the right of an up step.

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LargestAreaSequenceEntry(path)The largest entry of the area sequence: the greatest area to the right of an up step.

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.

Definition

Epsil, as the implementation runs it.

on DyckPath, over _x

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

Examples

Basic #

the step word $UUDDUD$

live ↗

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

the step word $UDUUDD$

live ↗

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

Properties #

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

live ↗

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

Known value · FindStat St000442

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

live ↗

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

Known value · FindStat St000442

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

live ↗

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

Known value · FindStat St000442

About

Examples	5
Crosswalk	FindStat