UpStepsAfterLastDoubleRise

Up steps after the first step of the last double rise; all of them if there is none.

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UpStepsAfterLastDoubleRise(path)Up steps after the first step of the last double rise; all of them if there is none.

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) < 2 else Apply(
                                (m: number) => Count(Filter(
                                                             Range(
                                                                    1,
                                                                    Length(_x),
                                                                  ),
                                                             Function(
                                                                       _x[i] ==
                                                                       1 && i > m,
                                                                       i,
                                                                     ),
                                                           )),
                                Apply(
                                       (doubles: list<integer>) => 0 if Length(doubles) ==
                                                                        0 else Max(doubles),
                                       Filter(
                                               Range(1, Length(_x) - 1),
                                               Function(_x[i] == 1 && _x[i + 1] ==
                                                                      1, i),
                                             ),
                                     ),
                              )

Examples

Basic #

the step word $UUDDUD$

live ↗

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

the step word $UDUUDD$

live ↗

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

Properties #

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

live ↗

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

Known value · FindStat St000678

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

live ↗

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

Known value · FindStat St000678

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

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{DyckPath}}([1, 0, 1, 0, 1, 0]), \text{UpStepsAfterLastDoubleRise})$Out$3$

Known value · FindStat St000678

About

Examples	5
Crosswalk	FindStat