Crossings

SetPartition: pairs of arcs a < b < c < d with a~c and b~d. Permutation: pairs i, j with i < j <= p(i) < p(j) or p(i) < p(j) < i < j.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000042via PerfectMatching✓↗St000039via Permutation✓↗St000232via SetPartition✓↗

Crossings(p)SetPartition: pairs of arcs a < b < c < d with a~c and b~d. Permutation: pairs i, j with i < j <= p(i) < p(j) or p(i) < p(j) < i < j.

Typed signatures

Type	Package	Applies
(permutation | 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.
• Read off the standard arc representation: within each block, consecutive elements are linked, and a crossing is two arcs whose spans interleave rather than nest or sit apart.
• Defined over Permutation as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
• Takes a Permutation 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.
• Crossings of the arcs of the permutation diagram (Corteel).

Definition

Epsil, as the implementation runs it.

on Permutation, over _x

0 if Length(_x) < 1 else Sum(Map(Function(Count(Filter(
                                                        Range(1, Length(_x)),
                                                        Function(i < j && j <=
                                                                          _x[i] && _x[i] <
                                                                              _x[j] ||
                                                                 _x[i] < _x[j] &&
                                                                 _x[j] < i && i < j, j),
                                                      )), i), Range(
                                                                    1,
                                                                    Length(_x),
                                                                  )))

on SetPartition, over _x

0 if Length(Flatten(Map(Function(Most(b), b), _x))) < 2 else Sum(Map(
                                                                      Function(Count(Filter(
                                                                                             Range(
                                                                                                    i + 1,
                                                                                                    Length(Flatten(Map(Function(Most(b), b), _x))),
                                                                                                    1,
                                                                                                  ),
                                                                                             Function([Max(
                                                                                                            Flatten(Map(Function(Most(b), b), _x))[i],
                                                                                                            Flatten(Map(Function(Most(b), b), _x))[j],
                                                                                                          ) <
                                                                                                       (Flatten(Map(Function(Rest(b), b), _x))[i] if Flatten(Map(Function(Most(b), b), _x))[i] <
                                                                                                                                                     Flatten(Map(Function(Most(b), b), _x))[j] else Flatten(Map(Function(Rest(b), b), _x))[j]) &&
                                                                                                       (Flatten(Map(Function(Rest(b), b), _x))[i] if Flatten(Map(Function(Most(b), b), _x))[i] <
                                                                                                                                                     Flatten(Map(Function(Most(b), b), _x))[j] else Flatten(Map(Function(Rest(b), b), _x))[j]) <
                                                                                                       (Flatten(Map(Function(Rest(b), b), _x))[j] if Flatten(Map(Function(Most(b), b), _x))[i] <
                                                                                                                                                     Flatten(Map(Function(Most(b), b), _x))[j] else Flatten(Map(Function(Rest(b), b), _x))[i]), Max(
                                                                                                              Flatten(Map(Function(Most(b), b), _x))[i],
                                                                                                              Flatten(Map(Function(Most(b), b), _x))[j],
                                                                                                            ) <
                                                                                                         (Flatten(Map(Function(Rest(b), b), _x))[i] if Flatten(Map(Function(Most(b), b), _x))[i] <
                                                                                                                                                       Flatten(Map(Function(Most(b), b), _x))[j] else Flatten(Map(Function(Rest(b), b), _x))[j]) &&
                                                                                                         (Flatten(Map(Function(Rest(b), b), _x))[j] if Flatten(Map(Function(Most(b), b), _x))[i] <
                                                                                                                                                       Flatten(Map(Function(Most(b), b), _x))[j] else Flatten(Map(Function(Rest(b), b), _x))[i]) <
                                                                                                         (Flatten(Map(Function(Rest(b), b), _x))[i] if Flatten(Map(Function(Most(b), b), _x))[i] <
                                                                                                                                                       Flatten(Map(Function(Most(b), b), _x))[j] else Flatten(Map(Function(Rest(b), b), _x))[j])][1], j),
                                                                                           )), i),
                                                                      Range(
                                                                             1,
                                                                             Length(Flatten(Map(Function(Most(b), b), _x))),
                                                                             1,
                                                                           ),
                                                                    ))

Examples

Basic #

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

live ↗

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

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

live ↗

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

the one-line word $312$

live ↗

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

the one-line word $2413$

live ↗

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

Properties #

FindStat's value for {{1}}

live ↗

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

Known value · FindStat St000232

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

live ↗

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

Known value · FindStat St000232

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

live ↗

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

Known value · FindStat St000232

FindStat's value for [1]

live ↗

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

Known value · FindStat St000039

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St000039

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St000039

About

Examples	10
Checked against	SageMath 5 agree
Crosswalk	FindStat ×3