UnitIntervalGraphTwinVertices

The twin vertices of the path's unit interval graph: the indices where the next up step and the next down step each follow directly.

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UnitIntervalGraphTwinVertices(path)The twin vertices of the path's unit interval graph: the indices where the next up step and the next down step each follow directly.

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.
• FindStat's St001189, the simple modules of dominant and codominant dimension zero in the Nakayama algebra. Two consecutive vertices are twins when they have the same closed neighborhood.

Definition

Epsil, as the implementation runs it.

on DyckPath, over _x

0 if Length(_x) < 4 else Apply((ups: list<integer>) => Apply(
                                                              (downs: list<integer>) => Count(Filter(
                                                                                                      Range(1, Length(_x) / 2 -
                                                                                                               1),
                                                                                                      Function(ups[i + 1] ==
                                                                                                               ups[i] + 1 &&
                                                                                                               downs[i + 1] ==
                                                                                                               downs[i] + 1, i),
                                                                                                    )),
                                                              Filter(
                                                                      Range(
                                                                             1,
                                                                             Length(_x),
                                                                           ),
                                                                      Function(_x[i] ==
                                                                               0, i),
                                                                    ),
                                                            ), Filter(
                                                                      Range(
                                                                             1,
                                                                             Length(_x),
                                                                           ),
                                                                      Function(_x[i] ==
                                                                               1, i),
                                                                    ))

Examples

Basic #

the step word $UUDDUD$

live ↗

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

the step word $UDUUDD$

live ↗

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

Properties #

FindStat's value for [1,0]

live ↗

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

Known value · FindStat St001189

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

live ↗

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

Known value · FindStat St001189

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

live ↗

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

Known value · FindStat St001189

About

Examples	5
Crosswalk	FindStat