CubeRootCount

The permutations whose cube is a fixed permutation of this cycle type: the product over part sizes d, with a parts of size d, of the number of ways to take the cube root of each part of size d alone or of three together as a cycle of length 3d.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001364via IntegerPartition✓↗

CubeRootCount(partition)The permutations whose cube is a fixed permutation of this cycle type: the product over part sizes d, with a parts of size d, of the number of ways to take the cube root of each part of size d alone or of three together as a cycle of length 3d.

Typed signatures

Type	Package	Applies
(integer_partition) -> number	combinatorics	always

Domain: Partition statistics

Details

• Defined over IntegerPartition as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
• Takes a IntegerPartition 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 IntegerPartition, over _x

1 if Length(_x) < 1 else Product(Map(Function(Sum(Map(
                                                       Function((1 if Count(Filter(
                                                                                    Range(1, Length(_x)),
                                                                                    Function(_x[j] == _x[i], j),
                                                                                  )) ==
                                                                      0 else Product(Range(
                                                                                           1,
                                                                                           Count(Filter(
                                                                                                         Range(1, Length(_x)),
                                                                                                         Function(_x[j] == _x[i], j),
                                                                                                       )),
                                                                                         ))) /
                                                                ((1 if Count(Filter(
                                                                                     Range(1, Length(_x)),
                                                                                     Function(_x[j] == _x[i], j),
                                                                                   )) -
                                                                       3s == 0 else Product(Range(1, Count(Filter(
                                                                                                                 Range(1, Length(_x)),
                                                                                                                 Function(_x[j] == _x[i], j),
                                                                                                               )) -
                                                                                                   3s))) *
                                                                 6 ^ s *
                                                                 (1 if s == 0 else Product(Range(1, s)))) * (2 * _x[i] *
                                                                    _x[i]) ^ s if _x[i] %
                                                                             3 >
                                                                             0 ||
                                                                             Count(Filter(
                                                                                           Range(1, Length(_x)),
                                                                                           Function(_x[j] == _x[i], j),
                                                                                         )) ==
                                                                             3s else 0, s),
                                                       Range(0, (Count(Filter(
                                                                               Range(1, Length(_x)),
                                                                               Function(_x[j] == _x[i], j),
                                                                             )) -
                                                                 Count(Filter(
                                                                               Range(1, Length(_x)),
                                                                               Function(_x[j] == _x[i], j),
                                                                             )) %
                                                                 3) / 3, 1),
                                                     )) if i == Length(_x) || _x[i] >
                                                                             _x[i + 1] else 1, i), Range(1, Length(_x))))

Examples

Basic #

the partition $4 + 2 + 1$ of $7$

live ↗

In$\mathrm{CubeRootCount}({\operatorname{IntegerPartition}}([4, 2, 1]))$Out$1$

the partition $3 + 3 + 1$ of $7$

live ↗

In$\mathrm{CubeRootCount}({\operatorname{IntegerPartition}}([3, 3, 1]))$Out$0$

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St001364

FindStat's value for [2]

live ↗

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

Known value · FindStat St001364

FindStat's value for [3]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerPartition}}([3]), \text{CubeRootCount})$Out$0$

Known value · FindStat St001364

About

Examples	5
Crosswalk	FindStat