ParallelogramPolyominoUpperInnerCorners

The inner corners of the polyomino's upper path: its turns from east to north, not counting the first.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001037via DyckPath✓↗

ParallelogramPolyominoUpperInnerCorners(path)The inner corners of the polyomino's upper path: its turns from east to north, not counting the first.

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.
• The upper path of FindStat's polyomino is the one that opens with an east step here, up to reflecting the polyomino in the diagonal.

Definition

Epsil, as the implementation runs it.

on DyckPath, over _x

0 if Length(_x) < 1 else Apply(
                                (east: list<integer>) => Sum(Map(
                                                                  Function(
                                                                            (1 if i == 1 else 1 if east[i - 1] ==
                                                                                                   1 else 0) if east[i] == 0 else 0,
                                                                            i,
                                                                          ),
                                                                  Range(
                                                                         1,
                                                                         Length(_x) /
                                                                         2 + 1,
                                                                       ),
                                                                )) - 1,
                                Join([0], Map(
                                               Function(1 - _x[2i], i),
                                               Range(1, Length(_x) / 2),
                                             )),
                              )

Examples

Basic #

the step word $UUDDUD$

live ↗

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

the step word $UDUUDD$

live ↗

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

Properties #

FindStat's value for [1,0]

live ↗

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

Known value · FindStat St001037

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

live ↗

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

Known value · FindStat St001037

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

live ↗

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

Known value · FindStat St001037

About

Examples	5
Crosswalk	FindStat