HillsOfSizeThree

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

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001141via DyckPath✓↗

HillsOfSizeThree(path)Occurrences of three up steps then three 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) < 6 else Count(Filter(
                                       Range(1, Length(_x) - 5),
                                       Function(
                                                 _x[i + 0] == 1 && _x[i + 1] ==
                                                                   1 && _x[i + 2] ==
                                                                       1 && _x[i +
                                                                              3] ==
                                                                           0 && _x[i + 4] ==
                                                                               0 &&
                                                 _x[i + 5] == 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$\mathrm{HillsOfSizeThree}({\operatorname{DyckPath}}([1, 1, 0, 0, 1, 0]))$Out$0$

the step word $UDUUDD$

live ↗

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

Properties #

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

live ↗

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

Known value · FindStat St001141

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

live ↗

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

Known value · FindStat St001141

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

live ↗

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

Known value · FindStat St001141

About

Examples	5
Crosswalk	FindStat