HillsOfSizeTwo

Occurrences of two up steps then two down steps, starting on the axis.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001139via DyckPath✓↗

HillsOfSizeTwo(path)Occurrences of two up steps then two down steps, starting on the axis.

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) < 4 else Count(Filter(
                                       Range(1, Length(_x) - 3),
                                       Function(_x[i + 0] == 1 && _x[i + 1] == 1 &&
                                                _x[i + 2] == 0 && _x[i + 3] == 0 && (0 if i ==
                                                                            1 else Sum(Map(
                                                                                           Function(2 * _x[k] -
                                                                                                    1, k),
                                                                                           Range(1, i - 1),
                                                                                         ))) ==
                                                                      0, i),
                                     ))

Examples

Basic #

the step word $UUDDUD$

live ↗

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

the step word $UDUUDD$

live ↗

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

Properties #

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

live ↗

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

Known value · FindStat St001139

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

live ↗

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

Known value · FindStat St001139

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

live ↗

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

Known value · FindStat St001139

About

Examples	5
Crosswalk	FindStat