PeakHeightProduct

The product of the heights of the peaks.

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PeakHeightProduct(path)The product of the heights of the peaks.

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

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

Examples

Basic #

the step word $UUDDUD$

live ↗

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

the step word $UDUUDD$

live ↗

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

Properties #

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

live ↗

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

Known value · FindStat St001959

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

live ↗

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

Known value · FindStat St001959

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

live ↗

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

Known value · FindStat St001959

About

Examples	5
Crosswalk	FindStat