TerminalClosers

Closers e with every larger element also a closer.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000838via PerfectMatching✓↗St001050via SetPartition✓↗

TerminalClosers(partition)Closers e with every larger element also a closer.

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 Count(Filter(
                                       Map(Function(Last(b), b), _x),
                                       Function(Count(Filter(
                                                              Map(Function(
                                                                            Last(b),
                                                                            b,
                                                                          ), _x),
                                                              Function(c > e, c),
                                                            )) == Max(Map(
                                                                          Function(Last(b), b),
                                                                          _x,
                                                                        )) - e, e),
                                     ))

Examples

Basic #

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

live ↗

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

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

live ↗

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

Properties #

FindStat's value for {{1}}

live ↗

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

Known value · FindStat St001050

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

live ↗

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

Known value · FindStat St001050

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

live ↗

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

Known value · FindStat St001050

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Examples	5
Crosswalk	FindStat ×2