EvenOddCrank

The largest even part minus the number of odd parts.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000997via IntegerPartition✓↗

EvenOddCrank(partition)The largest even part minus the number of odd parts.

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.
• Not the Andrews-Garvan Crank: this is the largest even part minus the number of odd parts.

Definition

Epsil, as the implementation runs it.

on IntegerPartition, over _x

0 if Length(_x) < 1 else Max(Map(
                                  Function(_x[i] if _x[i] % 2 == 0 else 0, i),
                                  Range(1, Length(_x)),
                                )) - Count(Filter(
                                                  Range(1, Length(_x)),
                                                  Function(_x[i] % 2 == 1, i),
                                                ))

Examples

Basic #

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

live ↗

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

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

live ↗

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

Properties #

FindStat's value for [2]

live ↗

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

Known value · FindStat St000997

FindStat's value for [3]

live ↗

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

Known value · FindStat St000997

FindStat's value for [4]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerPartition}}([4]), \text{EvenOddCrank})$Out$4$

Known value · FindStat St000997

About

Examples	5
Crosswalk	FindStat