ConjugateOddParts

Odd parts of the conjugate — equivalently, the distinct part sizes of λ.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000992via IntegerPartition✓↗

ConjugateOddParts(partition)Odd parts of the conjugate — equivalently, the distinct part sizes of λ.

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 Count(Filter(Map(
                                           Function(Count(Filter(
                                                                  Range(
                                                                         1,
                                                                         Length(_x),
                                                                       ),
                                                                  Function(_x[j] >=
                                                                           i, j),
                                                                )), i),
                                           Range(1, Max(_x)),
                                         ), Function(i % 2 == 1, i)))

Examples

Basic #

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

live ↗

In$\mathrm{ConjugateOddParts}({\operatorname{IntegerPartition}}([4, 2, 1]))$Out$3$

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

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St000992

FindStat's value for [2]

live ↗

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

Known value · FindStat St000992

FindStat's value for [3]

live ↗

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

Known value · FindStat St000992

About

Examples	5
Crosswalk	FindStat