BargraphHalfPerimeter

Up steps plus horizontal steps of the bargraph.

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BargraphHalfPerimeter(composition)Up steps plus horizontal steps of the bargraph.

Typed signatures

Type	Package	Applies
(composition) -> number	combinatorics	always

Domain: Composition statistics

Details

• Defined over IntegerComposition as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
• Takes a IntegerComposition 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.
• Deutsch and Elizalde, A bijection between bargraphs and Dyck paths, and FindStat call it the semiperimeter.

Definition

Epsil, as the implementation runs it.

on IntegerComposition, over _x

0 if Length(_x) < 1 else Length(_x) + _x[1] +
                         (0 if Length(_x) < 2 else Sum(Map(Function(
                                                                     _x[i + 1] -
                                                                     _x[i] if _x[i +
                                                                            1] >
                                                                         _x[i] else 0,
                                                                     i,
                                                                   ), Range(
                                                                            1,
                                                                            Length(_x) -
                                                                            1,
                                                                          ))))

Examples

Basic #

the composition $[2,1,3,1,1]$

live ↗

In$\mathrm{BargraphHalfPerimeter}({\operatorname{IntegerComposition}}([2, 1, 3, 1, 1]))$Out$9$

Properties #

FindStat's value for [2]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerComposition}}([2]), \text{BargraphHalfPerimeter})$Out$3$

Known value · FindStat St000806

FindStat's value for [3]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerComposition}}([3]), \text{BargraphHalfPerimeter})$Out$4$

Known value · FindStat St000806

FindStat's value for [4]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerComposition}}([4]), \text{BargraphHalfPerimeter})$Out$5$

Known value · FindStat St000806

About

Examples	4
Crosswalk	FindStat