ClosersBelowLargestOpener

The pairs whose larger point is below the largest opener, the larger point of the pair opening last.

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ClosersBelowLargestOpener(partition)The pairs whose larger point is below the largest opener, the larger point of the pair opening last.

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 St000840. Every other opener is below the largest, so the closers below it number the largest opener less the pairs.

Definition

Epsil, as the implementation runs it.

on PerfectMatching, over _x

0 if Length(_x) < 1 else Max(Map(Function(b[1], b), _x)) - Length(_x)

Examples

Basic #

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

live ↗

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

Properties #

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

live ↗

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

Known value · FindStat St000840

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

live ↗

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

Known value · FindStat St000840

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

live ↗

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

Known value · FindStat St000840

About

Examples	4
Crosswalk	FindStat