DistinctSubsetSums

Distinct sums of non-empty sub-multisets of the parts.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000548via IntegerPartition✓↗

DistinctSubsetSums(partition)Distinct sums of non-empty 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.

Definition

Epsil, as the implementation runs it.

on IntegerPartition, over _x

0 if Length(_x) < 1 else Apply(
                                (s: collection) => Length(s) - 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$\mathrm{DistinctSubsetSums}({\operatorname{IntegerPartition}}([4, 2, 1]))$Out$7$

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

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St000548

FindStat's value for [2]

live ↗

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

Known value · FindStat St000548

FindStat's value for [3]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerPartition}}([3]), \text{DistinctSubsetSums})$Out$1$

Known value · FindStat St000548

About

Examples	5
Crosswalk	FindStat