SecondAlladiWeight

Half the last part (rounded up if odd), times the product over consecutive parts of half the gap less 1 for each even one of the two.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000478via IntegerPartition✓↗

SecondAlladiWeight(partition)Half the last part (rounded up if odd), times the product over consecutive parts of half the gap less 1 for each even one of the two.

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.
• Alladi's second weight: for each part, delta is 1 if it is even and 0 if odd; the factor for the last part is (last + 1 - delta)/2 and for each consecutive pair (gap - delta - delta')/2.

Definition

Epsil, as the implementation runs it.

on IntegerPartition, over _x

0 if Length(_x) < 1 else (_x[Length(_x)] + 1 -
                          (1 if _x[Length(_x)] % 2 == 0 else 0)) / 2 if Length(_x) == 1 else 0 if Count(Filter(
                                                                           Range(1, Length(_x) - 1),
                                                                           Function(_x[i] -
                                                                                    _x[i + 1] -
                                                                                    (1 if _x[i] % 2 ==
                                                                                          0 else 0) -
                                                                                    (1 if _x[i + 1] %
                                                                                          2 ==
                                                                                          0 else 0) ==
                                                                                    0, i),
                                                                         )) > 0 else (_x[Length(_x)] + 1 -
                                                                                     (1 if _x[Length(_x)] % 2 ==
                                                                                           0 else 0)) /
                                                                                    2 *
                                                                                    Product(Map(
                                                                                                 Function((_x[i] -
                                                                                                           _x[i + 1] -
                                                                                                           (1 if _x[i] % 2 ==
                                                                                                                 0 else 0) -
                                                                                                           (1 if _x[i + 1] %
                                                                                                                 2 ==
                                                                                                                 0 else 0)) /
                                                                                                          2, i),
                                                                                                 Range(1, Length(_x) - 1),
                                                                                               ))

Examples

Basic #

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

live ↗

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

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

live ↗

In$\mathrm{SecondAlladiWeight}({\operatorname{IntegerPartition}}([3, 3, 1]))$Out$0$

Properties #

FindStat's value for [2]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerPartition}}([2]), \text{SecondAlladiWeight})$Out$1$

Known value · FindStat St000478

FindStat's value for [3]

live ↗

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

Known value · FindStat St000478

FindStat's value for [4]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerPartition}}([4]), \text{SecondAlladiWeight})$Out$2$

Known value · FindStat St000478

About

Examples	5
Crosswalk	FindStat