DistinctEvenSubsetSums

Distinct nonzero even sums of sub-multisets of the parts.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000547via IntegerPartition✓↗

DistinctEvenSubsetSums(partition)Distinct nonzero even sums of sub-multisets of the 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.
• Follows FindStat's Sage code and its data, not its title: the "partial sums" it counts are the sums of arbitrary sub-multisets of the parts (the code runs over all subsets), not of the running prefixes. The empty sum, 0, is not counted.

Definition

Epsil, as the implementation runs it.

on IntegerPartition, over _x

0 if Length(_x) < 1 else Apply(
                                (s: list<integer>) => Count(Filter(
                                                                    s,
                                                                    Function(i %
                                                                             2 ==
                                                                             0, i),
                                                                  )) - 1,
                                Fold(Function(Unique(Join(
                                                           a,
                                                           Map(Function(
                                                                         i + b,
                                                                         i,
                                                                       ), a),
                                                         )), a, b), [0], _x),
                              )

Examples

Basic #

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

live ↗

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

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

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St000547

FindStat's value for [2]

live ↗

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

Known value · FindStat St000547

FindStat's value for [3]

live ↗

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

Known value · FindStat St000547

About

Examples	5
Crosswalk	FindStat