DominatedPartitionCount

The partitions of the same size that this one dominates (partial sums of the parts at least those of the other), itself included.

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DominatedPartitionCount(partition)The partitions of the same size that this one dominates (partial sums of the parts at least those of the other), 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 Apply(
                                (w: list<integer>) => Sum(w),
                                Fold(
                                      Function(Map(Function(Map(
                                                                 Function(Sum(Map(
                                                                                   Function(a[r - c +
                                                                                              1][d], d),
                                                                                   Range(c, Sum(_x) + 1, 1),
                                                                                 )) if c <= r &&
                                                                                      r - 1 <=
                                                                                      Sum(Map(
                                                                                               Function(_x[q] if q <= Length(_x) else 0, q),
                                                                                               Range(1, b, 1),
                                                                                             )) else 0, c),
                                                                 Range(
                                                                        1,
                                                                        Sum(_x) +
                                                                        1,
                                                                      ),
                                                               ), r), Range(
                                                                            1,
                                                                            Sum(_x) +
                                                                            1,
                                                                          )), a, b),
                                      Map(Function(Map(
                                                        Function(
                                                                  1 if r == 1 &&
                                                                       c == Sum(_x) +
                                                                            1 else 0,
                                                                  c,
                                                                ),
                                                        Range(1, Sum(_x) + 1),
                                                      ), r), Range(
                                                                   1,
                                                                   Sum(_x) + 1,
                                                                 )),
                                      Range(1, Sum(_x)),
                                    )[Sum(_x) + 1],
                              )

Examples

Basic #

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

live ↗

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

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

live ↗

In$\mathrm{DominatedPartitionCount}({\operatorname{IntegerPartition}}([3, 3, 1]))$Out$8$

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St000321

FindStat's value for [2]

live ↗

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

Known value · FindStat St000321

FindStat's value for [3]

live ↗

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

Known value · FindStat St000321

About

Examples	5
Crosswalk	FindStat