AlladiWeight

The last part times the product over consecutive parts of the gap less 1.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000477via IntegerPartition✓↗

AlladiWeight(partition)The last part times the product over consecutive parts of the gap less 1.

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 _x[Length(_x)] if Length(_x) == 1 else 0 if Count(Filter(
                                                                                   Range(1, Length(_x) - 1),
                                                                                   Function(_x[i] -
                                                                                            _x[i + 1] ==
                                                                                            1, i),
                                                                                 )) >
                                                                     0 else _x[Length(_x)] *
                                                                           Product(Map(
                                                                                        Function(_x[i] -
                                                                                                 _x[i + 1] -
                                                                                                 1, i),
                                                                                        Range(1, Length(_x) - 1),
                                                                                      ))

Examples

Basic #

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

live ↗

In$\mathrm{AlladiWeight}({\operatorname{IntegerPartition}}([4, 2, 1]))$Out$0$

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

live ↗

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

Properties #

FindStat's value for [2]

live ↗

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

Known value · FindStat St000477

FindStat's value for [3]

live ↗

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

Known value · FindStat St000477

FindStat's value for [4]

live ↗

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

Known value · FindStat St000477

About

Examples	5
Crosswalk	FindStat