InvariantGraphCount

The simple graphs on the points that a permutation of this cycle type maps to themselves: 2 to the number of edge orbits.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000514via IntegerPartition✓↗

InvariantGraphCount(partition)The simple graphs on the points that a permutation of this cycle type maps to themselves: 2 to the number of edge orbits.

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 2 ^ (Sum(Map(
                                       Function((_x[i] - _x[i] % 2) / 2, i),
                                       Range(1, Length(_x)),
                                     )) + Sum(Map(
                                                  Function(Sum(Map(
                                                                    Function(
                                                                              _x[i] *
                                                                              _x[j] /
                                                                              LCM(_x[i], _x[j]) if j < i else 0,
                                                                              j,
                                                                            ),
                                                                    Range(
                                                                           1,
                                                                           Length(_x),
                                                                         ),
                                                                  )), i),
                                                  Range(1, Length(_x)),
                                                )))

Examples

Basic #

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

live ↗

In$\mathrm{InvariantGraphCount}({\operatorname{IntegerPartition}}([4, 2, 1]))$Out$128$

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

live ↗

In$\mathrm{InvariantGraphCount}({\operatorname{IntegerPartition}}([3, 3, 1]))$Out$128$

Properties #

FindStat's value for [2]

live ↗

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

Known value · FindStat St000514

FindStat's value for [3]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerPartition}}([3]), \text{InvariantGraphCount})$Out$2$

Known value · FindStat St000514

FindStat's value for [4]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerPartition}}([4]), \text{InvariantGraphCount})$Out$4$

Known value · FindStat St000514

About

Examples	5
Crosswalk	FindStat