OccurrencesOf1Over2Over3Openers3Closers12

Occurrences of the pattern 1/2/3 with 3 the least element of its block and 1 and 2 the greatest elements of their blocks.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000583via SetPartition✓↗

OccurrencesOf1Over2Over3Openers3Closers12(partition)Occurrences of the pattern 1/2/3 with 3 the least element of its block and 1 and 2 the greatest elements of their blocks.

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

Examples

Basic #

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

live ↗

In$\mathrm{OccurrencesOf1Over2Over3Openers3Closers12}({\operatorname{SetPartition}}([[1, 3], [2]]))$Out$0$

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

live ↗

In$\mathrm{OccurrencesOf1Over2Over3Openers3Closers12}({\operatorname{SetPartition}}([[1], [2, 4], [3]]))$Out$0$

Properties #

FindStat's value for {{1}}

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{SetPartition}}([[1]]), \text{OccurrencesOf1Over2Over3Openers3Closers12})$Out$0$

Known value · FindStat St000583

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

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{SetPartition}}([[1, 2]]), \text{OccurrencesOf1Over2Over3Openers3Closers12})$Out$0$

Known value · FindStat St000583

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

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{SetPartition}}([[1, 2, 3]]), \text{OccurrencesOf1Over2Over3Openers3Closers12})$Out$0$

Known value · FindStat St000583

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Examples	5
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