MinimalSubpartitionSizeDifference

The smallest difference in size between the two halves of a split of the parts into two sub-multisets.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000835via IntegerPartition✓↗

MinimalSubpartitionSizeDifference(partition)The smallest difference in size between the two halves of a split of the parts into two sub-multisets.

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 Apply(
                                (s: list<integer>) => Min(Map(
                                                               Function(i * 2 -
                                                                        Sum(_x), i),
                                                               Filter(
                                                                       s,
                                                                       Function(i * 2 >=
                                                                                Sum(_x), i),
                                                                     ),
                                                             )),
                                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$\mathrm{MinimalSubpartitionSizeDifference}({\operatorname{IntegerPartition}}([4, 2, 1]))$Out$1$

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

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St000835

FindStat's value for [2]

live ↗

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

Known value · FindStat St000835

FindStat's value for [3]

live ↗

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

Known value · FindStat St000835

About

Examples	5
Crosswalk	FindStat