SubsetGreatestCommonDivisorSum

The sum, over the non-empty subsets of the parts, of their greatest common divisor: the sum over d of phi(d) times one less than 2 to the number of parts divisible by d.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001985via IntegerPartition✓↗

SubsetGreatestCommonDivisorSum(partition)The sum, over the non-empty subsets of the parts, of their greatest common divisor: the sum over d of phi(d) times one less than 2 to the number of parts divisible by d.

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

0 if Length(_x) < 1 else Sum(Map(Function(Count(Filter(
                                                        Range(1, d),
                                                        Function(k * d /
                                                                 LCM(k, d) == 1, k),
                                                      )) *
                                          (2 ^ Count(Filter(
                                                             Range(
                                                                    1,
                                                                    Length(_x),
                                                                  ),
                                                             Function(_x[i] % d ==
                                                                      0, i),
                                                           )) - 1), d), Range(1, Max(_x))))

Examples

Basic #

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

live ↗

In$\mathrm{SubsetGreatestCommonDivisorSum}({\operatorname{IntegerPartition}}([4, 2, 1]))$Out$12$

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

live ↗

In$\mathrm{SubsetGreatestCommonDivisorSum}({\operatorname{IntegerPartition}}([3, 3, 1]))$Out$13$

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St001985

FindStat's value for [2]

live ↗

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

Known value · FindStat St001985

FindStat's value for [3]

live ↗

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

Known value · FindStat St001985

About

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Crosswalk	FindStat