IsSelfConjugate

1 when λ equals its conjugate, 0 otherwise.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001122via IntegerPartition✓↗

IsSelfConjugate(partition)1 when λ equals its conjugate, 0 otherwise.

SageMathenumeratio_stat("IntegerPartition", partition, "IsSelfConjugate")

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

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

Examples

Basic #

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

live ↗

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

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

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St001122

FindStat's value for [2]

live ↗

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

Known value · FindStat St001122

FindStat's value for [3]

live ↗

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

Known value · FindStat St001122

About

Examples	5
Checked against	SageMath 5 agree
Crosswalk	FindStatSageMath

Implementation

mappedsagepackages/oracle/src/mappings.ts