WeightedSize

The sum over the cells of their row, counting rows from 0.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000185via IntegerPartition✓↗St000336via StandardTableau✓↗

WeightedSize(partition)The sum of the parts weighted by their row, counting rows from 0.

SageMathenumeratio_stat("IntegerPartition", partition, "WeightedSize")

WeightedSize(tableau)The sum of the leg lengths of the cells, the sum over the cells of their row counted from 0.

SageMathenumeratio_stat("IntegerPartition", tableau, "WeightedSize")

Typed signatures

Type	Package	Applies
(integer_partition | standard_tableau) -> number	combinatorics	always

Domain: Partition and standard tableau 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.
• Defined over StandardTableau as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
• Takes a StandardTableau 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.
• FindStat's leg major index (St000336), a statistic of the shape; on a partition it is the same head. The empty tableau gives 0.

Definition

Epsil, as the implementation runs it.

on IntegerPartition, over _x

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

on StandardTableau, over _x

0 if Length(_x) < 1 else Apply(
                                (lengths: list<integer>) => Max(Sum(Map(
                                                                         Function((i - 1) *
                                                                                  lengths[i], i),
                                                                         Range(1, Length(_x)),
                                                                       )), 0),
                                Map(Function(Length(r), r), _x),
                              )

Examples

Basic #

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

live ↗

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

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

live ↗

In$\mathrm{WeightedSize}({\operatorname{IntegerPartition}}([3, 3, 1]))$Out$5$

the standard tableau $[[1,2,4],[3,6],[5]]$

live ↗

In$\mathrm{WeightedSize}({\operatorname{StandardTableau}}([[1, 2, 4], [3, 6], [5]]))$Out$4$

Properties #

FindStat's value for [1]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerPartition}}([1]), \text{WeightedSize})$Out$0$

Known value · FindStat St000185

FindStat's value for [2]

live ↗

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

Known value · FindStat St000185

FindStat's value for [3]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerPartition}}([3]), \text{WeightedSize})$Out$0$

Known value · FindStat St000185

FindStat's value for [[1]]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{StandardTableau}}([[1]]), \text{WeightedSize})$Out$0$

Known value · FindStat St000336

FindStat's value for [[1,2]]

live ↗

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

Known value · FindStat St000336

FindStat's value for [[1,2,3]]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{StandardTableau}}([[1, 2, 3]]), \text{WeightedSize})$Out$0$

Known value · FindStat St000336

About

Examples	9
Checked against	SageMath 5 agree
Crosswalk	FindStat ×2SageMath

Implementation

mappedsagepackages/oracle/src/mappings.ts