SumOfHookLengths

The total of all hook lengths.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaHook length formula↗MathWorldHookLengthFormula↗WikidataQ17163411↗FindStatSt000869via IntegerPartition✓↗

SumOfHookLengths(partition)The total of all hook lengths.

SageMathenumeratio_stat("IntegerPartition", partition, "SumOfHookLengths")

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.
• Hook of cell (i, j) is arm + leg + 1 = (λ_i - j) + (λ'_j - i) + 1.

Definition

Epsil, as the implementation runs it.

on IntegerPartition, over _x

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

Examples

Basic #

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

live ↗

In$\mathrm{SumOfHookLengths}({\operatorname{IntegerPartition}}([4, 2, 1]))$Out$18$

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

live ↗

In$\mathrm{SumOfHookLengths}({\operatorname{IntegerPartition}}([3, 3, 1]))$Out$18$

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St000869

FindStat's value for [2]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerPartition}}([2]), \text{SumOfHookLengths})$Out$3$

Known value · FindStat St000869

FindStat's value for [3]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerPartition}}([3]), \text{SumOfHookLengths})$Out$6$

Known value · FindStat St000869

About

Examples	5
Checked against	SageMath 5 agree
Crosswalk	WikipediaMathWorldWikidataFindStatSageMath

Implementation

mappedsagepackages/oracle/src/mappings.ts