PeakRectangleAreaSum

The sum, over consecutive peaks, of the area of the rectangle they span with the valley between them.

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PeakRectangleAreaSum(path)The sum, over consecutive peaks, of the area of the rectangle they span with the valley between them.

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.
• Positions are counted from 1; only differences enter.

Definition

Epsil, as the implementation runs it.

on DyckPath, over _x

0 if Length(_x) < 2 else Apply(
                                (peaks: list<integer>) => Apply(
                                                                 (valleys: list<integer>) => 0 if Length(valleys) == 0 else Sum(Map(Function((valleys[a] - peaks[a]) *
                                                                                                                                             (peaks[a + 1] -
                                                                                                                                              valleys[a]), a), Range(1, Length(valleys)))),
                                                                 Filter(
                                                                         Range(
                                                                                1,
                                                                                Length(_x) - 1,
                                                                              ),
                                                                         Function(_x[i] == 0 &&
                                                                                  _x[i + 1] ==
                                                                                  1, i),
                                                                       ),
                                                               ),
                                Filter(
                                        Range(1, Length(_x) - 1),
                                        Function(
                                                  _x[i] == 1 && _x[i + 1] == 0,
                                                  i,
                                                ),
                                      ),
                              )

Examples

Basic #

the step word $UUDDUD$

live ↗

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

the step word $UDUUDD$

live ↗

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

Properties #

FindStat's value for [1,0]

live ↗

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

Known value · FindStat St000645

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

live ↗

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

Known value · FindStat St000645

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

live ↗

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

Known value · FindStat St000645

About

Examples	5
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