IdentityMatchingEdgeCount

Pairs {2k - 1, 2k}: the edges shared with the matching 12, 34, 56, ....

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IdentityMatchingEdgeCount(partition)Pairs {2k - 1, 2k}: the edges shared with the matching 12, 34, 56, ....

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 St001997. The identity matching is 12, 34, 56, ....

Definition

Epsil, as the implementation runs it.

on PerfectMatching, over _x

Count(Filter(_x, Function(b[1] % 2 == 1 && Last(b) - b[1] == 1, b)))

Examples

Basic #

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

live ↗

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

Properties #

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

live ↗

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

Known value · FindStat St001997

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

live ↗

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

Known value · FindStat St001997

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

live ↗

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

Known value · FindStat St001997

About

Examples	4
Crosswalk	FindStat