DownStepMajorIndex

The major index counting positions in down steps only: over each valley, the number of down steps up to and including its down step; summed.

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DownStepMajorIndex(path)The major index counting positions in down steps only: over each valley, the number of down steps up to and including its down step; summed.

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 Sum(Map(
                                  Function(
                                            Count(Filter(
                                                          Range(1, i),
                                                          Function(
                                                                    _x[d] == 0,
                                                                    d,
                                                                  ),
                                                        )),
                                            i,
                                          ),
                                  Filter(
                                          Range(1, Length(_x) - 1),
                                          Function(
                                                    _x[i] == 0 && _x[i + 1] == 1,
                                                    i,
                                                  ),
                                        ),
                                ))

Examples

Basic #

the step word $UUDDUD$

live ↗

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

the step word $UDUUDD$

live ↗

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

Properties #

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

live ↗

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

Known value · FindStat St000947

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

live ↗

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

Known value · FindStat St000947

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

live ↗

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

Known value · FindStat St000947

About

Examples	5
Crosswalk	FindStat