OccurrencesOf12Over34

Occurrences of the pattern 12/34: a < b < c < d with a, b in one block and c, d in another.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000560via SetPartition✓↗

OccurrencesOf12Over34(partition)Occurrences of the pattern 12/34: a < b < c < d with a, b in one block and c, d in another.

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(Flatten(_x)) < 4 else Sum(Map(
                                           Function(Sum(Map(
                                                             Function(Sum(Map(
                                                                               Function(Count(Filter(
                                                                                                      Range(c + 1, Length(Flatten(_x))),
                                                                                                      Function(Sum(Map(
                                                                                                                        Function(q if Contains(_x[q], a) else 0, q),
                                                                                                                        Range(1, Length(_x)),
                                                                                                                      )) ==
                                                                                                               Sum(Map(
                                                                                                                        Function(q if Contains(_x[q], b) else 0, q),
                                                                                                                        Range(1, Length(_x)),
                                                                                                                      )) &&
                                                                                                               Sum(Map(
                                                                                                                        Function(q if Contains(_x[q], c) else 0, q),
                                                                                                                        Range(1, Length(_x)),
                                                                                                                      )) ==
                                                                                                               Sum(Map(
                                                                                                                        Function(q if Contains(_x[q], d) else 0, q),
                                                                                                                        Range(1, Length(_x)),
                                                                                                                      )) &&
                                                                                                               Abs(Sum(Map(
                                                                                                                            Function(q if Contains(_x[q], a) else 0, q),
                                                                                                                            Range(1, Length(_x)),
                                                                                                                          )) -
                                                                                                                   Sum(Map(
                                                                                                                            Function(q if Contains(_x[q], c) else 0, q),
                                                                                                                            Range(1, Length(_x)),
                                                                                                                          ))) >
                                                                                                               0, d),
                                                                                                    )), c),
                                                                               Range(b + 1, Length(Flatten(_x)) - 1),
                                                                             )), b),
                                                             Range(
                                                                    a + 1,
                                                                    Length(Flatten(_x)) -
                                                                    2,
                                                                  ),
                                                           )), a),
                                           Range(1, Length(Flatten(_x)) - 3),
                                         ))

Examples

Basic #

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

live ↗

In$\mathrm{OccurrencesOf12Over34}({\operatorname{SetPartition}}([[1, 3], [2]]))$Out$0$

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

live ↗

In$\mathrm{OccurrencesOf12Over34}({\operatorname{SetPartition}}([[1], [2, 4], [3]]))$Out$0$

Properties #

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

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{SetPartition}}([[1, 2]]), \text{OccurrencesOf12Over34})$Out$0$

Known value · FindStat St000560

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

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{SetPartition}}([[1, 2, 3]]), \text{OccurrencesOf12Over34})$Out$0$

Known value · FindStat St000560

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

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{SetPartition}}([[1], [2]]), \text{OccurrencesOf12Over34})$Out$0$

Known value · FindStat St000560

About

Examples	5
Crosswalk	FindStat