AlignedPairCount

Pairs of pairs (a, b), (c, d) with a < b < c < d.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000719via PerfectMatching✓↗

AlignedPairCount(partition)Pairs of pairs (a, b), (c, d) with a < b < c < d.

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 St000719, its alignments. Any two pairs cross, nest or align, so the three counts sum to n choose 2.

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(Sum(Map(Function(Sum(Map(
                                                                                                                            Function(1 if closers[i] < openers[j] else 0, j),
                                                                                                                            Filter(
                                                                                                                                    Range(1, Length(_x)),
                                                                                                                                    Function(j > i, j),
                                                                                                                                  ),
                                                                                                                          )), i), Range(1, Length(_x)))), 0),
                                                                  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$\mathrm{AlignedPairCount}({\operatorname{PerfectMatching}}([[1, 4], [2, 3], [5, 6]]))$Out$2$

Properties #

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

live ↗

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

Known value · FindStat St000719

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

live ↗

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

Known value · FindStat St000719

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

live ↗

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

Known value · FindStat St000719

About

Examples	4
Crosswalk	FindStat