DistinctBlockPairs

Pairs of elements in different blocks.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000564via SetPartition✓↗

DistinctBlockPairs(partition)Pairs of elements in different blocks.

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) < 1 else Sum(Map(Function(Length(b), b), _x)) * (Sum(Map(
                                                                          Function(Length(b), b),
                                                                          _x,
                                                                        )) - 1) /
                         2 - (0 if Length(_x) < 1 else Sum(Map(
                                                               Function(Length(b) *
                                                                        (Length(b) -
                                                                         1) / 2, b),
                                                               _x,
                                                             )))

Examples

Basic #

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

live ↗

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

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

live ↗

In$\mathrm{DistinctBlockPairs}({\operatorname{SetPartition}}([[1], [2, 4], [3]]))$Out$5$

Properties #

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

live ↗

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

Known value · FindStat St000564

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

live ↗

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

Known value · FindStat St000564

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

live ↗

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

Known value · FindStat St000564

About

Examples	5
Crosswalk	FindStat