InternalPoints

Elements with an earlier and a later element whose blocks open after theirs does.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000562via SetPartition✓↗

InternalPoints(partition)Elements with an earlier and a later element whose blocks open after theirs does.

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.
• An element e is internal when some f < e < g have blocks with larger least element than e's block (Mansour, Section 5.5).

Definition

Epsil, as the implementation runs it.

on SetPartition, over _x

0 if Sum(Map(Function(Length(b), b), _x)) < 3 else Count(Filter(
                                                                 Range(
                                                                        2,
                                                                        Sum(Map(
                                                                                 Function(Length(b), b),
                                                                                 _x,
                                                                               )) -
                                                                        1,
                                                                      ),
                                                                 Function(Max(Map(
                                                                                   Function(Map(
                                                                                                 Function(Filter(_x, Function(Contains(b, e), b))[1][1], e),
                                                                                                 Range(1, Sum(Map(Function(Length(b), b), _x))),
                                                                                               )[j], j),
                                                                                   Range(1, e - 1),
                                                                                 )) >
                                                                          Map(
                                                                               Function(
                                                                                         Filter(_x, Function(Contains(b, e), b))[1][1],
                                                                                         e,
                                                                                       ),
                                                                               Range(
                                                                                      1,
                                                                                      Sum(Map(Function(Length(b), b), _x)),
                                                                                    ),
                                                                             )[e] && Max(Map(
                                                                                       Function(Map(
                                                                                                     Function(Filter(_x, Function(Contains(b, e), b))[1][1], e),
                                                                                                     Range(1, Sum(Map(Function(Length(b), b), _x))),
                                                                                                   )[j], j),
                                                                                       Range(
                                                                                              e + 1,
                                                                                              Sum(Map(Function(Length(b), b), _x)),
                                                                                            ),
                                                                                     )) >
                                                                              Map(
                                                                                   Function(
                                                                                             Filter(_x, Function(Contains(b, e), b))[1][1],
                                                                                             e,
                                                                                           ),
                                                                                   Range(1, Sum(Map(Function(Length(b), b), _x))),
                                                                                 )[e], e),
                                                               ))

Examples

Basic #

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

live ↗

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

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

live ↗

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

Properties #

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

live ↗

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

Known value · FindStat St000562

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

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{SetPartition}}([[1, 2, 3]]), \text{InternalPoints})$Out$0$

Known value · FindStat St000562

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

live ↗

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

Known value · FindStat St000562

About

Examples	5
Crosswalk	FindStat