BargraphValleyHeightSum

The sum of the heights of the valleys of the bargraph.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000807via IntegerComposition✓↗

BargraphValleyHeightSum(composition)The sum of the heights of the valleys 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.
• A valley is a fall, level steps, then a rise (Deutsch and Elizalde, A bijection between bargraphs and Dyck paths).

Definition

Epsil, as the implementation runs it.

on IntegerComposition, over _x

0 if Length(_x) < 1 else 0 if Length(_x) < 2 else Fold(
                                                        Function(
                                                                  [1, a[2]] if _x[b] <
                                                                               _x[b - 1] else [0, a[2] +
                                                                                         a[1] *
                                                                                         _x[b - 1]] if _x[b] >
                                                                                              _x[b - 1] else a,
                                                                  a,
                                                                  b,
                                                                ),
                                                        [0, 0],
                                                        Range(2, Length(_x), 1),
                                                      )[2]

Examples

Basic #

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

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St000807

FindStat's value for [2]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerComposition}}([2]), \text{BargraphValleyHeightSum})$Out$0$

Known value · FindStat St000807

FindStat's value for [3]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerComposition}}([3]), \text{BargraphValleyHeightSum})$Out$0$

Known value · FindStat St000807

About

Examples	4
Crosswalk	FindStat