RefinementCount

The partitions that refine this one, whose parts subdivide to give theirs. Itself included.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000345via IntegerPartition✓↗

RefinementCount(partition)The partitions that refine this one, whose parts subdivide to give theirs. Itself included.

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) < 2 else Length(Fold(
                                                            Function(Unique(Join(
                                                                                  a,
                                                                                  Map(
                                                                                       Function(Apply(
                                                                                                       (mc: integer) => Apply(
                                                                                                                               (sj: integer) => Apply(
                                                                                                                                                       (sa: integer) => mc -
                                                                                                                                                                        ((Sum(_x) + 1) ^
                                                                                                                                                                         sj -
                                                                                                                                                                         ((Sum(_x) + 1) ^
                                                                                                                                                                          sa +
                                                                                                                                                                          (Sum(_x) + 1) ^
                                                                                                                                                                          (sj - sa))) if 2sa <= sj &&
                                                                                                                                                                            (mc -
                                                                                                                                                                             mc %
                                                                                                                                                                             (Sum(_x) + 1) ^
                                                                                                                                                                             sj) /
                                                                                                                                                                            (Sum(_x) + 1) ^
                                                                                                                                                                            sj %
                                                                                                                                                                            (Sum(_x) + 1) >=
                                                                                                                                                                            1 else mc,
                                                                                                                                                       1 +
                                                                                                                                                       (e - 1) %
                                                                                                                                                       (Sum(_x) - 1) ^
                                                                                                                                                       2 %
                                                                                                                                                       (Sum(_x) - 1),
                                                                                                                                                     ),
                                                                                                                               2 +
                                                                                                                               ((e - 1) %
                                                                                                                                (Sum(_x) - 1) ^
                                                                                                                                2 -
                                                                                                                                (e - 1) %
                                                                                                                                (Sum(_x) - 1) ^
                                                                                                                                2 %
                                                                                                                                (Sum(_x) - 1)) /
                                                                                                                               (Sum(_x) - 1),
                                                                                                                             ),
                                                                                                       a[1 +
                                                                                                         (e - 1 -
                                                                                                          (e - 1) %
                                                                                                          (Sum(_x) - 1) ^
                                                                                                          2) /
                                                                                                         (Sum(_x) - 1) ^
                                                                                                         2],
                                                                                                     ), e),
                                                                                       Range(1, Length(a) *
                                                                                                (Sum(_x) - 1) ^
                                                                                                2),
                                                                                     ),
                                                                                )), a, b),
                                                            [Sum(Map(
                                                                      Function((Sum(_x) + 1) ^
                                                                               _x[i], i),
                                                                      Range(
                                                                             1,
                                                                             Length(_x),
                                                                           ),
                                                                    ))],
                                                            Range(
                                                                   1,
                                                                   Sum(_x) - 1,
                                                                   1,
                                                                 ),
                                                          ))

Examples

Basic #

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

live ↗

In$\mathrm{RefinementCount}({\operatorname{IntegerPartition}}([4, 2, 1]))$Out$8$

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

live ↗

In$\mathrm{RefinementCount}({\operatorname{IntegerPartition}}([3, 3, 1]))$Out$6$

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St000345

FindStat's value for [2]

live ↗

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

Known value · FindStat St000345

FindStat's value for [3]

live ↗

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

Known value · FindStat St000345

About

Examples	5
Crosswalk	FindStat