MaximalRightCrossingCount

The most pairs (c, d) with a < c < b < d, over the pairs (a, b): arcs that cross a given arc at its right end.

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MaximalRightCrossingCount(partition)The most pairs (c, d) with a < c < b < d, over the pairs (a, b): arcs that cross a given arc at its right end.

Typed signatures

Type	Package	Applies
(perfect_matching) -> number	combinatorics	always

Domain: Perfect matching statistics

Details

• Defined over PerfectMatching as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
• Takes a PerfectMatching 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 St001047 counts these, though its title says arcs crossing a given arc: its code sweeps the closers and counts the arcs opened inside and still open, which leaves out the arcs crossing at the left end.

Definition

Epsil, as the implementation runs it.

on PerfectMatching, over _x

0 if Length(_x) < 1 else Apply((openers: list<integer>) => Apply(
                                                                  (closers: list<integer>) => Max(Map(
                                                                                                       Function(Count(Filter(
                                                                                                                              Range(1, Length(_x)),
                                                                                                                              Function(openers[i] < openers[j] &&
                                                                                                                                       openers[j] < closers[i] &&
                                                                                                                                       closers[i] < closers[j], j),
                                                                                                                            )), i),
                                                                                                       Range(1, Length(_x)),
                                                                                                     )),
                                                                  Map(
                                                                       Function(Last(b), b),
                                                                       _x,
                                                                     ),
                                                                ), Map(
                                                                       Function(b[1], b),
                                                                       _x,
                                                                     ))

Examples

Basic #

the matching $\{1,4\} \mid \{2,3\} \mid \{5,6\}$

live ↗

In$\mathrm{MaximalRightCrossingCount}({\operatorname{PerfectMatching}}([[1, 4], [2, 3], [5, 6]]))$Out$0$

Properties #

FindStat's value for [(1,2)]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{PerfectMatching}}([[1, 2]]), \text{MaximalRightCrossingCount})$Out$0$

Known value · FindStat St001047

FindStat's value for [(1,2),(3,4)]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{PerfectMatching}}([[1, 2], [3, 4]]), \text{MaximalRightCrossingCount})$Out$0$

Known value · FindStat St001047

FindStat's value for [(1,3),(2,4)]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{PerfectMatching}}([[1, 3], [2, 4]]), \text{MaximalRightCrossingCount})$Out$1$

Known value · FindStat St001047

About

Examples	4
Crosswalk	FindStat