Crank

The Andrews-Garvan crank: the largest part when λ has no 1s, else (parts larger than the number of 1s) minus (the number of 1s).

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaCrank of a partition↗WikidataQ5182182↗FindStatSt000474via IntegerPartition✓↗

Crank(partition)The Andrews-Garvan crank: the largest part when λ has no 1s, else (parts larger than the number of 1s) minus (the number of 1s).

SageMathenumeratio_stat("IntegerPartition", partition, "Crank")

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 Max(_x) if Count(Filter(
                                                  Range(1, Length(_x)),
                                                  Function(_x[i] == 1, i),
                                                )) == 0 else Count(Filter(
                                                                          Range(1, Length(_x)),
                                                                          Function(_x[i] >
                                                                                   Count(Filter(
                                                                                                 Range(1, Length(_x)),
                                                                                                 Function(_x[i] == 1, i),
                                                                                               )), i),
                                                                        )) -
                                                            Count(Filter(
                                                                          Range(1, Length(_x)),
                                                                          Function(
                                                                                    _x[i] == 1,
                                                                                    i,
                                                                                  ),
                                                                        ))

Examples

Basic #

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

live ↗

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

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

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St000474

FindStat's value for [2]

live ↗

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

Known value · FindStat St000474

FindStat's value for [3]

live ↗

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

Known value · FindStat St000474

About

Examples	5
Checked against	SageMath 5 agree
Crosswalk	WikipediaWikidataFindStatSageMath

Implementation

mappedsagepackages/oracle/src/mappings.ts