IntertwiningNumber

Pairs a < b in different blocks with no element of either block between them.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000490via SetPartition✓↗

IntertwiningNumber(partition)Pairs a < b in different blocks with no element of either block between them.

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.
• The block intertwiners of Ehrenborg and Readdy (Juggling and applications to q-analogues), counted over all pairs of blocks.

Definition

Epsil, as the implementation runs it.

on SetPartition, over _x

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

Examples

Basic #

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

live ↗

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

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

live ↗

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

Properties #

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

live ↗

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

Known value · FindStat St000490

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

live ↗

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

Known value · FindStat St000490

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

live ↗

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

Known value · FindStat St000490

About

Examples	5
Crosswalk	FindStat