Antisingletons

Elements i with i + 1 in the same block, counting n + 1 as 1.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000248via SetPartition✓↗

Antisingletons(partition)Elements i with i + 1 in the same block, counting n + 1 as 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) < 1 else (0 if Length(Flatten(Map(Function(Most(b), b), _x))) <
                               1 else Count(Filter(
                                                   Range(
                                                          1,
                                                          Length(Flatten(Map(
                                                                              Function(Most(b), b),
                                                                              _x,
                                                                            ))),
                                                          1,
                                                        ),
                                                   Function(Flatten(Map(
                                                                         Function(Rest(b), b),
                                                                         _x,
                                                                       ))[i] -
                                                            Flatten(Map(
                                                                         Function(Most(b), b),
                                                                         _x,
                                                                       ))[i] == 1, i),
                                                 ))) + (1 if Last(_x[1]) ==
                                                            Sum(Map(
                                                                     Function(Length(b), b),
                                                                     _x,
                                                                   )) else 0)

Examples

Basic #

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

live ↗

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

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

live ↗

In$\mathrm{Antisingletons}({\operatorname{SetPartition}}([[1], [2, 4], [3]]))$Out$0$

Properties #

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

live ↗

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

Known value · FindStat St000248

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

live ↗

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

Known value · FindStat St000248

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

live ↗

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

Known value · FindStat St000248

About

Examples	5
Crosswalk	FindStat