SubpartitionCount

The partitions contained in this one.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000108via IntegerPartition✓↗

SubpartitionCount(partition)The partitions contained in this one: the choices of a row size for each row, at most the row's size and the row above's.

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 Max(Sum(Fold(
                                       Function(Map(
                                                     Function(Sum(Map(
                                                                       Function(a[k], k),
                                                                       Range(
                                                                              i,
                                                                              Max(_x) +
                                                                              1,
                                                                            ),
                                                                     )) if i >= 1 &&
                                                                          i <= b +
                                                                               1 else 0, i),
                                                     Range(1, Max(_x) + 1),
                                                   ), a, b),
                                       Map(
                                            Function(1 if i >= 1 else 0, i),
                                            Range(1, Max(_x) + 1),
                                          ),
                                       Rest(_x),
                                     )), 0)

Examples

Basic #

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

live ↗

In$\mathrm{SubpartitionCount}({\operatorname{IntegerPartition}}([4, 2, 1]))$Out$19$

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

live ↗

In$\mathrm{SubpartitionCount}({\operatorname{IntegerPartition}}([3, 3, 1]))$Out$16$

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St000108

FindStat's value for [2]

live ↗

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

Known value · FindStat St000108

FindStat's value for [3]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerPartition}}([3]), \text{SubpartitionCount})$Out$4$

Known value · FindStat St000108

About

Examples	5
Crosswalk	FindStat