DistinctMultiplicities

How many different multiplicities the part sizes have.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000905via IntegerComposition✓↗St000897via IntegerPartition✓↗

DistinctMultiplicities(partition)How many different multiplicities the part sizes have.

Typed signatures

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

on IntegerComposition, over _x

0 if Length(_x) < 1 else Length(Unique(Map(
                                            Function(Count(Filter(
                                                                   Range(
                                                                          1,
                                                                          Length(_x),
                                                                        ),
                                                                   Function(_x[j] ==
                                                                            _x[i], j),
                                                                 )), i),
                                            Range(1, Length(_x)),
                                          )))

Examples

Basic #

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

live ↗

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

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

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St000897

FindStat's value for [2]

live ↗

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

Known value · FindStat St000897

FindStat's value for [3]

live ↗

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

Known value · FindStat St000897

About

Examples	5
Crosswalk	FindStat ×2