FlooredHalfMultiplicitySum

The sum, over part sizes, of half the multiplicity rounded down.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000150via IntegerPartition✓↗

FlooredHalfMultiplicitySum(partition)The sum, over part sizes, of half the multiplicity rounded down.

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 Sum(Map(Function((Count(Filter(
                                                         Range(1, Length(_x)),
                                                         Function(_x[i] == k, i),
                                                       )) - Count(Filter(
                                                                         Range(1, Length(_x)),
                                                                         Function(
                                                                                   _x[i] == k,
                                                                                   i,
                                                                                 ),
                                                                       )) % 2) /
                                          2, k), Range(1, Max(_x))))

Examples

Basic #

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

live ↗

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

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

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St000150

FindStat's value for [2]

live ↗

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

Known value · FindStat St000150

FindStat's value for [3]

live ↗

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

Known value · FindStat St000150

About

Examples	5
Crosswalk	FindStat