MultipartitionCount

Tuples of partitions whose sizes are the parts: the product of the partition numbers of the parts.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000933via IntegerPartition✓↗

MultipartitionCount(partition)The number of tuples of partitions whose sizes are the parts: the product of the partition numbers 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 Apply(
                                (p: list<integer>) => Product(Map(
                                                                   Function(
                                                                             p[_x[i] +
                                                                               1],
                                                                             i,
                                                                           ),
                                                                   Range(
                                                                          1,
                                                                          Length(_x),
                                                                        ),
                                                                 )),
                                Fold(
                                      Function(Map(
                                                    Function(Sum(Map(
                                                                      Function(a[i -
                                                                                 t * b], t),
                                                                      Range(
                                                                             0,
                                                                             (i -
                                                                              1 -
                                                                              (i -
                                                                               1) %
                                                                              b) /
                                                                             b,
                                                                           ),
                                                                    )), i),
                                                    Range(1, Max(_x) + 1),
                                                  ), a, b),
                                      Map(
                                           Function(1 if i == 1 else 0, i),
                                           Range(1, Max(_x) + 1),
                                         ),
                                      Range(1, Max(_x)),
                                    ),
                              )

Examples

Basic #

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

live ↗

In$\mathrm{MultipartitionCount}({\operatorname{IntegerPartition}}([4, 2, 1]))$Out$10$

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

live ↗

In$\mathrm{MultipartitionCount}({\operatorname{IntegerPartition}}([3, 3, 1]))$Out$9$

Properties #

FindStat's value for [2]

live ↗

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

Known value · FindStat St000933

FindStat's value for [3]

live ↗

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

Known value · FindStat St000933

FindStat's value for [4]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerPartition}}([4]), \text{MultipartitionCount})$Out$5$

Known value · FindStat St000933

About

Examples	5
Crosswalk	FindStat