FirstComponentSemilength

The position of the first return to height 0, counted in up steps: half the steps of the first primitive component.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000026via DyckPath✓↗

FirstComponentSemilength(path)The position of the first return to height 0, counted in up steps: half the steps of the first primitive component.

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.
• FindStat's "position of the first return" counts up steps, not steps: UD is 1.

Definition

Epsil, as the implementation runs it.

on DyckPath, over _x

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

Examples

Basic #

the step word $UUDDUD$

live ↗

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

the step word $UDUUDD$

live ↗

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

Properties #

FindStat's value for [1,0]

live ↗

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

Known value · FindStat St000026

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

live ↗

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

Known value · FindStat St000026

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

live ↗

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

Known value · FindStat St000026

About

Examples	5
Crosswalk	FindStat