RightOpenerSmaller

Pairs of an opener j and a larger element i of an earlier block.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000491via SetPartition✓↗

RightOpenerSmaller(partition)Pairs of an opener j and a larger element i of an earlier 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.
• FindStat's ros (right, opener, smaller): an inversion of the set partition in Steingrímsson's sense.

Definition

Epsil, as the implementation runs it.

on SetPartition, over _x

0 if Length(_x) < 2 else Sum(Map(Function(Count(Filter(
                                                        Range(
                                                               1,
                                                               Length(Flatten(_x)),
                                                             ),
                                                        Function(Sum(Map(
                                                                          Function(
                                                                                    q if Contains(_x[q], i) else 0,
                                                                                    q,
                                                                                  ),
                                                                          Range(1, Length(_x)),
                                                                        )) < k &&
                                                                 i > _x[k][1], i),
                                                      )), k), Range(
                                                                    1,
                                                                    Length(_x),
                                                                  )))

Examples

Basic #

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

live ↗

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

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

live ↗

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

Properties #

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

live ↗

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

Known value · FindStat St000491

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

live ↗

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

Known value · FindStat St000491

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

live ↗

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

Known value · FindStat St000491

About

Examples	5
Crosswalk	FindStat