SetPartitionsWithBlockSizesCount

The number of set partitions whose block sizes are the parts: the size factorial over the product of the factorials of the parts and of the multiplicities.

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SetPartitionsWithBlockSizesCount(partition)The number of set partitions whose block sizes are the parts: the size factorial over the product of the factorials of the parts and of the multiplicities.

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 Sum(_x) == 0 else Product(Range(1, Sum(_x)))) /
                         (Product(Map(Function(
                                                1 if _x[i] == 0 else Product(Range(1, _x[i])),
                                                i,
                                              ), Range(1, Length(_x)))) *
                          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$\mathrm{SetPartitionsWithBlockSizesCount}({\operatorname{IntegerPartition}}([4, 2, 1]))$Out$105$

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

live ↗

In$\mathrm{SetPartitionsWithBlockSizesCount}({\operatorname{IntegerPartition}}([3, 3, 1]))$Out$70$

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St000049

FindStat's value for [2]

live ↗

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

Known value · FindStat St000049

FindStat's value for [3]

live ↗

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

Known value · FindStat St000049

About

Examples	5
Crosswalk	FindStat