RisesOfLengthOneAtOddHeight

Rises of length 1 that start at an odd height.

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RisesOfLengthOneAtOddHeight(path)Rises of length 1 that start at an odd height.

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] == 0 && _x[i + 1] == 1 &&
                                                 _x[i + 2] == 0 && Sum(Map(
                                                                            Function(2 * _x[k] -
                                                                                     1, k),
                                                                            Range(1, i),
                                                                          )) % 2 ==
                                                                   1,
                                                 i,
                                               ),
                                     ))

Examples

Basic #

the step word $UUDDUD$

live ↗

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

the step word $UDUUDD$

live ↗

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

Properties #

FindStat's value for [1,0]

live ↗

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

Known value · FindStat St001172

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

live ↗

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

Known value · FindStat St001172

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

live ↗

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

Known value · FindStat St001172

About

Examples	5
Crosswalk	FindStat