LargestPart

The largest part.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000381via IntegerComposition✓↗St000147via IntegerPartition✓↗St001447via PlanePartition✓↗

LargestPart(partition)The largest part.

SageMathenumeratio_stat("IntegerPartition", partition, "LargestPart")

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 Max(_x)

on IntegerComposition, over _x

0 if Length(_x) < 1 else Max(_x)

on PlanePartition, over _x

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

Examples

Basic #

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

live ↗

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

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

live ↗

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

the plane partition with rows $32 \mid 21 \mid 1$: the corner entry

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{PlanePartition}}([[3, 2], [2, 1], [1]]), \text{LargestPart})$Out$3$

the weak composition $[0,2,1]$

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{WeakComposition}}([0, 2, 1]), \text{LargestPart})$Out$2$

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St000147

FindStat's value for [2]

live ↗

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

Known value · FindStat St000147

FindStat's value for [3]

live ↗

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

Known value · FindStat St000147

About

Examples	7
Checked against	SageMath 5 agree
Crosswalk	FindStat ×3SageMath

Implementation

mappedsagepackages/oracle/src/mappings.ts