BerkovichGarvanRank

The sum of a checkerboard filling of the diagram, +1 on cells whose coordinates sum to an even number and -1 on the others.

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BerkovichGarvanRank(partition)The sum of a checkerboard filling of the diagram, +1 on cells whose coordinates sum to an even number and -1 on the others.

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.
• A row of even length cancels and one of odd length leaves a single cell, counted + in the first row, - in the second, and so on.

Definition

Epsil, as the implementation runs it.

on IntegerPartition, over _x

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

Examples

Basic #

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

live ↗

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

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

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St001383

FindStat's value for [2]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerPartition}}([2]), \text{BerkovichGarvanRank})$Out$0$

Known value · FindStat St001383

FindStat's value for [3]

live ↗

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

Known value · FindStat St001383

About

Examples	5
Crosswalk	FindStat