HalfMacMahonEqualIndex

Half the sum of the positions of double rises and double falls.

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HalfMacMahonEqualIndex(path)Half the sum of the positions of double rises and double falls.

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.
• Half of MacMahonEqualIndex, which is even on a Dyck path.

Definition

Epsil, as the implementation runs it.

on DyckPath, over _x

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

Examples

Basic #

the step word $UUDDUD$

live ↗

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

the step word $UDUUDD$

live ↗

In$\mathrm{HalfMacMahonEqualIndex}({\operatorname{DyckPath}}([1, 0, 1, 1, 0, 0]))$Out$4$

Properties #

FindStat's value for [1,0]

live ↗

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

Known value · FindStat St000979

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

live ↗

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

Known value · FindStat St000979

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

live ↗

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

Known value · FindStat St000979

About

Examples	5
Crosswalk	FindStat