RisesOfLengthAtLeastThree

Rises (maximal runs of up steps) of length at least 3.

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RisesOfLengthAtLeastThree(path)Rises (maximal runs of up steps) of length at least 3.

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(
                                (lengths: list<integer>) => Count(Filter(
                                                                          lengths,
                                                                          Function(r >= 3, r),
                                                                        )),
                                Apply(
                                       (u: list<integer>) => Map(
                                                                  Function(
                                                                            u[1] if k == 1 else u[k] -
                                                                                                u[k - 1],
                                                                            k,
                                                                          ),
                                                                  Range(
                                                                         1,
                                                                         Length(u),
                                                                       ),
                                                                ),
                                       Map(
                                            Function(Count(Filter(
                                                                   Range(1, i),
                                                                   Function(_x[k] ==
                                                                            1, k),
                                                                 )), i),
                                            Filter(
                                                    Range(1, Length(_x) - 1),
                                                    Function(_x[i] == 1 && _x[i +
                                                                              1] ==
                                                                           0, i),
                                                  ),
                                          ),
                                     ),
                              )

Examples

Basic #

the step word $UUDDUD$

live ↗

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

the step word $UDUUDD$

live ↗

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

Properties #

FindStat's value for [1,0]

live ↗

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

Known value · FindStat St000660

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

live ↗

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

Known value · FindStat St000660

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

live ↗

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

Known value · FindStat St000660

About

Examples	5
Crosswalk	FindStat