DistinctParts

How many distinct part sizes occur.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaInteger partition↗MathWorldPartition↗WikidataQ1082910↗Britannicatopic/partition-of-an-integer↗OEISA000009↗FindStatSt000903via IntegerComposition✓↗St000159via IntegerPartition✓↗St001453via PlanePartition✓↗

DistinctParts(partition)How many distinct part sizes occur.

Typed signatures

Type	Package	Applies
(composition | integer_partition | plane_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 Count(Filter(
                                       Range(1, Max(_x)),
                                       Function(Count(Filter(
                                                              Range(
                                                                     1,
                                                                     Length(_x),
                                                                   ),
                                                              Function(_x[i] ==
                                                                       j, i),
                                                            )) > 0, j),
                                     ))

on IntegerComposition, over _x

Length(Unique(_x))

on PlanePartition, over _x

0 if Length(Flatten(_x)) < 1 else Apply((reading: list<integer>) => Apply(
                                                                           (lengths: list<integer>) => Max(Sum(Map(Function(1 if Max(Sum(Map(Function(1 if e == k else 0, e), reading)), 0) >
                                                                                                                                 0 else 0, k), Range(1, Max(reading), 1))), 0),
                                                                           Map(
                                                                                Function(Length(r), r),
                                                                                _x,
                                                                              ),
                                                                         ), Flatten(_x))

Examples

Basic #

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

live ↗

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

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

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St000159

FindStat's value for [2]

live ↗

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

Known value · FindStat St000159

FindStat's value for [3]

live ↗

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

Known value · FindStat St000159

About

Examples	5
Crosswalk	WikipediaMathWorldWikidataBritannicaOEISFindStat ×3