RunsOfLengthOne

Maximal runs of equal steps that are a single step.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001669via DyckPath✓↗

RunsOfLengthOne(path)Maximal runs of equal steps that are a single step.

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.
• Counts runs of both kinds of step, up and down; FindStat's title, "single rises", is misleading.

Definition

Epsil, as the implementation runs it.

on DyckPath, over _x

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

Examples

Basic #

the step word $UUDDUD$

live ↗

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

the step word $UDUUDD$

live ↗

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

Properties #

FindStat's value for [1,0]

live ↗

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

Known value · FindStat St001669

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

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{DyckPath}}([1, 0, 1, 0]), \text{RunsOfLengthOne})$Out$4$

Known value · FindStat St001669

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

live ↗

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

Known value · FindStat St001669

About

Examples	5
Crosswalk	FindStat