BlockMajorIndex

The sum over blocks i of i times the elements of block i above the least element of block i + 1.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000565via SetPartition✓↗

BlockMajorIndex(partition)The sum over blocks i of i times the elements of block i above the least element of block i + 1.

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(
                                           i * Count(Filter(
                                                             _x[i],
                                                             Function(e >
                                                                      _x[i + 1][1], e),
                                                           )),
                                           i,
                                         ), Range(1, Length(_x) - 1)))

Examples

Basic #

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

live ↗

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

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

live ↗

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

Properties #

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

live ↗

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

Known value · FindStat St000565

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

live ↗

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

Known value · FindStat St000565

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

live ↗

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

Known value · FindStat St000565

About

Examples	5
Crosswalk	FindStat