NonFinalConstantRunsOfLengthAtLeastTwo

Maximal runs of equal steps, at least two long, that are followed by a step of the other kind: the occurrences of two up steps then a down step, or two down steps then an up step.

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NonFinalConstantRunsOfLengthAtLeastTwo(path)Maximal runs of equal steps, at least two long, that are followed by a step of the other kind: the occurrences of two up steps then a down step, or two down steps then an up 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.

Definition

Epsil, as the implementation runs it.

on DyckPath, over _x

(0 if Length(_x) < 3 else Count(Filter(
                                        Range(1, Length(_x) - 2),
                                        Function(_x[i] == 1 && _x[i + 1] == 1 &&
                                                 _x[i + 2] == 0, i),
                                      ))) +
(0 if Length(_x) < 3 else Count(Filter(
                                        Range(1, Length(_x) - 2),
                                        Function(_x[i] == 0 && _x[i + 1] == 0 &&
                                                 _x[i + 2] == 1, i),
                                      )))

Examples

Basic #

the step word $UUDDUD$

live ↗

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

the step word $UDUUDD$

live ↗

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

Properties #

FindStat's value for [1,0]

live ↗

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

Known value · FindStat St000340

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

live ↗

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

Known value · FindStat St000340

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

live ↗

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

Known value · FindStat St000340

About

Examples	5
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