SmallestMultiplicity

The smallest number of parts of one size, among the sizes that occur.

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SmallestMultiplicity(composition)The smallest number of parts of one size, among the sizes that occur.

Typed signatures

Type	Package	Applies
(composition) -> number	combinatorics	always

Domain: Composition statistics

Details

• Defined over IntegerComposition as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
• Takes a IntegerComposition 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 IntegerComposition, over _x

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

Examples

Basic #

the composition $[2,1,3,1,1]$

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St000902

FindStat's value for [2]

live ↗

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

Known value · FindStat St000902

FindStat's value for [3]

live ↗

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

Known value · FindStat St000902

About

Examples	4
Crosswalk	FindStat