RotationOrbitSize

The number of distinct partitions rotation i -> i + 1 (mod n) makes of this one.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000543via BinaryWord✓↗St000820via IntegerComposition✓↗St000780via PerfectMatching✓↗St000163via SetPartition✓↗

RotationOrbitSize(partition)The number of distinct partitions rotation i -> i + 1 (mod n) makes of this one.

Typed signatures

Type	Package	Applies
(composition | 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(_x) < 1 else Min(Filter(
                                     Range(1, Length(Flatten(_x))),
                                     Function(Sum(Map(Function(Count(Filter(
                                                                             Range(1, Length(Flatten(_x))),
                                                                             Function(Abs((1 if Sum(Map(
                                                                                                         Function(q if Contains(_x[q], i) else 0, q),
                                                                                                         Range(1, Length(_x)),
                                                                                                       )) ==
                                                                                                Sum(Map(
                                                                                                         Function(q if Contains(_x[q], j) else 0, q),
                                                                                                         Range(1, Length(_x)),
                                                                                                       )) else 0) -
                                                                                          (1 if Sum(Map(
                                                                                                         Function(q if Contains(_x[q], (i + k -
                                                                                                                                        1) %
                                                                                                                                       Length(Flatten(_x)) +
                                                                                                                                       1) else 0, q),
                                                                                                         Range(1, Length(_x)),
                                                                                                       )) ==
                                                                                                Sum(Map(
                                                                                                         Function(q if Contains(_x[q], (j + k -
                                                                                                                                        1) %
                                                                                                                                       Length(Flatten(_x)) +
                                                                                                                                       1) else 0, q),
                                                                                                         Range(1, Length(_x)),
                                                                                                       )) else 0)) >
                                                                                      0, j),
                                                                           )), i), Range(1, Length(Flatten(_x))))) ==
                                              0, k),
                                   ))

on IntegerComposition, over _x

0 if Length(_x) < 1 else Min(Filter(
                                     Range(1, Length(_x)),
                                     Function(Count(Filter(
                                                            Range(1, Length(_x)),
                                                            Function(Abs(_x[i] -
                                                                         _x[(i +
                                                                             r -
                                                                             1) %
                                                                            Length(_x) + 1]) > 0, i),
                                                          )) == 0, r),
                                   ))

Examples

Basic #

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

live ↗

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

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

live ↗

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

Properties #

FindStat's value for {{1}}

live ↗

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

Known value · FindStat St000163

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

live ↗

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

Known value · FindStat St000163

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

live ↗

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

Known value · FindStat St000163

About

Examples	5
Crosswalk	FindStat ×4