MultiplicityFactorialProduct

The product of the factorials of the multiplicities of the parts.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000706via IntegerPartition✓↗

MultiplicityFactorialProduct(partition)The product of the factorials of the multiplicities of the parts.

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 Product(Map(
                                      Function(Count(Filter(
                                                             Range(1, i),
                                                             Function(_x[j] ==
                                                                      _x[i], j),
                                                           )), i),
                                      Range(1, Length(_x)),
                                    ))

Examples

Basic #

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

live ↗

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

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

live ↗

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

Properties #

FindStat's value for [2]

live ↗

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

Known value · FindStat St000706

FindStat's value for [3]

live ↗

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

Known value · FindStat St000706

FindStat's value for [4]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerPartition}}([4]), \text{MultiplicityFactorialProduct})$Out$1$

Known value · FindStat St000706

About

Examples	5
Crosswalk	FindStat