GlobalMaxima

The number of points of the path at the greatest height it reaches.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000617via DyckPath✓↗

GlobalMaxima(path)The number of points of the path at the greatest height it reaches.

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.
• The origin counts as a point, so the empty path has one; otherwise these are its highest peaks.

Definition

Epsil, as the implementation runs it.

on DyckPath, over _x

1 if Length(_x) == 0 else Apply(
                                 (h: number) => Count(Filter(
                                                              Range(
                                                                     1,
                                                                     Length(_x),
                                                                   ),
                                                              Function(Sum(Map(
                                                                                Function(2 * _x[k] -
                                                                                         1, k),
                                                                                Range(1, i),
                                                                              )) ==
                                                                       h, i),
                                                            )),
                                 Max(Map(
                                          Function(Sum(Map(
                                                            Function(2 * _x[k] -
                                                                     1, k),
                                                            Range(1, i),
                                                          )), i),
                                          Range(1, Length(_x)),
                                        )),
                               )

Examples

Basic #

the step word $UUDDUD$

live ↗

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

the step word $UDUUDD$

live ↗

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

Properties #

FindStat's value for [1,0]

live ↗

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

Known value · FindStat St000617

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

live ↗

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

Known value · FindStat St000617

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

live ↗

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

Known value · FindStat St000617

About

Examples	5
Crosswalk	FindStat