PairwiseGreatestCommonDivisorSum

The sum of the greatest common divisors of all pairs of parts.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001961via IntegerPartition✓↗

PairwiseGreatestCommonDivisorSum(partition)The sum of the greatest common divisors of all pairs of parts.

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

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

live ↗

In$\mathrm{PairwiseGreatestCommonDivisorSum}({\operatorname{IntegerPartition}}([3, 3, 1]))$Out$5$

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St001961

FindStat's value for [2]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerPartition}}([2]), \text{PairwiseGreatestCommonDivisorSum})$Out$0$

Known value · FindStat St001961

FindStat's value for [3]

live ↗

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

Known value · FindStat St001961

About

Examples	5
Crosswalk	FindStat