ParallelogramPolyominoInnerCorners

The inner corners of the parallelogram polyomino: the turns from east to north on its two bounding paths, not counting the corners every polyomino has.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001036via DyckPath✓↗

ParallelogramPolyominoInnerCorners(path)The inner corners of the parallelogram polyomino: the turns from east to north on its two bounding paths, not counting the corners every polyomino has.

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

Examples

Basic #

the step word $UUDDUD$

live ↗

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

the step word $UDUUDD$

live ↗

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

Properties #

FindStat's value for [1,0]

live ↗

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

Known value · FindStat St001036

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

live ↗

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

Known value · FindStat St001036

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

live ↗

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

Known value · FindStat St001036

About

Examples	5
Crosswalk	FindStat