SetPartitionDimension

The sum of the lengths of the arcs, taking for each block its largest element minus its smallest.

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SetPartitionDimension(partition)The sum of the lengths of the arcs, taking for each block its largest element minus its smallest.

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.
• FindStat's St000728; the arcs join consecutive elements of a block, so the lengths add up to the block's span.

Definition

Epsil, as the implementation runs it.

on SetPartition, over _x

0 if Length(_x) < 1 else Sum(Map(Function(Last(b) - b[1], b), _x))

Examples

Basic #

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

live ↗

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

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

live ↗

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

Properties #

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

live ↗

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

Known value · FindStat St000728

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

live ↗

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

Known value · FindStat St000728

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

live ↗

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

Known value · FindStat St000728

About

Examples	5
Crosswalk	FindStat