MiddleNonRunHeight

The height of the middle turning point: the middle one of the peaks and valleys, left to right.

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MiddleNonRunHeight(path)The height of the middle turning point: the middle one of the peaks and valleys, left to right.

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) < 2 else Apply(
                                (turns: list<integer>) => turns[(Length(turns) +
                                                                 1) / 2],
                                Map(Function(
                                              Sum(Map(
                                                       Function(
                                                                 2 * _x[k] - 1,
                                                                 k,
                                                               ),
                                                       Range(1, i),
                                                     )),
                                              i,
                                            ), Filter(
                                                      Range(1, Length(_x) - 1),
                                                      Function(
                                                                _x[i] +
                                                                _x[i + 1] == 1,
                                                                i,
                                                              ),
                                                    )),
                              )

Examples

Basic #

the step word $UUDDUD$

live ↗

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

the step word $UDUUDD$

live ↗

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

Properties #

FindStat's value for [1,0]

live ↗

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

Known value · FindStat St001910

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

live ↗

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

Known value · FindStat St001910

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

live ↗

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

Known value · FindStat St001910

About

Examples	5
Crosswalk	FindStat