MinimalArcLength

The length of the shortest arc; the number of elements when there is no arc.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000729via SetPartition✓↗

MinimalArcLength(partition)The length of the shortest arc; the number of elements when there is no arc.

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 St000729.

Definition

Epsil, as the implementation runs it.

on SetPartition, over _x

0 if Length(_x) < 1 else Length(Flatten(_x)) if Length(Flatten(Map(Function(
                                                                             Most(b),
                                                                             b,
                                                                           ), _x))) <
                                                1 else Min(Map(
                                                               Function(Min(Map(
                                                                                 Function(b[j + 1] -
                                                                                          b[j], j),
                                                                                 Range(1, Length(b) - 1, 1),
                                                                               )), b),
                                                               Filter(
                                                                       _x,
                                                                       Function(
                                                                                 Length(b) > 1,
                                                                                 b,
                                                                               ),
                                                                     ),
                                                             ))

Examples

Basic #

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

live ↗

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

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

live ↗

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

Properties #

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

live ↗

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

Known value · FindStat St000729

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

live ↗

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

Known value · FindStat St000729

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

live ↗

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

Known value · FindStat St000729

About

Examples	5
Crosswalk	FindStat