OverlappingBlockPairs

Pairs of blocks A, B with min A < min B < max A < max B.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000563via SetPartition✓↗

OverlappingBlockPairs(partition)Pairs of blocks A, B with min A < min B < max A < max B.

Typed signatures

Type	Package	Applies
(set_partition) -> number	combinatorics	always

Domain: Set partition statistics

Details

• Defined over SetPartition as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
• Takes a SetPartition 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.

Definition

Epsil, as the implementation runs it.

on SetPartition, over _x

0 if Length(_x) < 2 else Sum(Map(Function(
                                           Count(Filter(
                                                         Range(
                                                                i + 1,
                                                                Length(_x),
                                                                1,
                                                              ),
                                                         Function(Map(
                                                                       Function(b[1], b),
                                                                       _x,
                                                                     )[j] <
                                                                  Map(
                                                                       Function(Last(b), b),
                                                                       _x,
                                                                     )[i] && Map(
                                                                           Function(Last(b), b),
                                                                           _x,
                                                                         )[i] <
                                                                      Map(
                                                                           Function(Last(b), b),
                                                                           _x,
                                                                         )[j], j),
                                                       )),
                                           i,
                                         ), Range(1, Length(_x))))

Examples

Basic #

the partition $\{1,3\} \mid \{2\}$

live ↗

In$\mathrm{OverlappingBlockPairs}({\operatorname{SetPartition}}([[1, 3], [2]]))$Out$0$

the partition $\{1\} \mid \{2,4\} \mid \{3\}$

live ↗

In$\mathrm{OverlappingBlockPairs}({\operatorname{SetPartition}}([[1], [2, 4], [3]]))$Out$0$

Properties #

FindStat's value for {{1,2}}

live ↗

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

Known value · FindStat St000563

FindStat's value for {{1,2,3}}

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{SetPartition}}([[1, 2, 3]]), \text{OverlappingBlockPairs})$Out$0$

Known value · FindStat St000563

FindStat's value for {{1},{2}}

live ↗

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

Known value · FindStat St000563

About

Examples	5
Crosswalk	FindStat