FirstBlockSecondOrSmallestNonsingletonMaximum

The smaller of the second element of the block holding 1 and the smallest closer; 1 when {1} is a block.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001784via SetPartition✓↗

FirstBlockSecondOrSmallestNonsingletonMaximum(partition)The smaller of the second element of the block holding 1 and the smallest closer; 1 when {1} is a block.

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.
• A closer is the greatest element of a non-singleton block.

Definition

Epsil, as the implementation runs it.

on SetPartition, over _x

0 if Length(_x) < 1 else 1 if Length(_x[1]) == 1 else Min([
                                                            _x[1][2],
                                                            Min(Map(
                                                                     Function(Last(b), b),
                                                                     Filter(
                                                                             _x,
                                                                             Function(Length(b) > 1, b),
                                                                           ),
                                                                   )),
                                                          ])

Examples

Basic #

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

live ↗

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

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

live ↗

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

Properties #

FindStat's value for {{1}}

live ↗

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

Known value · FindStat St001784

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

live ↗

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

Known value · FindStat St001784

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

live ↗

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

Known value · FindStat St001784

About

Examples	5
Crosswalk	FindStat