AddableCells

Cells that can be added to the diagram and leave a partition: one more than the distinct parts.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000318via IntegerPartition✓↗

AddableCells(partition)Cells that can be added to the diagram and leave a partition: one more than the distinct parts.

SageMathenumeratio_stat("IntegerPartition", partition, "AddableCells")

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 Count(Filter(
                                       Range(1, Max(_x)),
                                       Function(Count(Filter(
                                                              Range(
                                                                     1,
                                                                     Length(_x),
                                                                   ),
                                                              Function(_x[i] ==
                                                                       j, i),
                                                            )) > 0, j),
                                     )) + 1

Examples

Basic #

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

live ↗

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

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

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St000318

FindStat's value for [2]

live ↗

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

Known value · FindStat St000318

FindStat's value for [3]

live ↗

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

Known value · FindStat St000318

About

Examples	5
Checked against	SageMath 5 agree
Crosswalk	FindStatSageMath

Implementation

mappedsagepackages/oracle/src/mappings.ts