LowerCoverCountInDominanceOrder

The partitions of the same size that this one covers in dominance order: the upper covers of its conjugate.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000480via IntegerPartition✓↗

LowerCoverCountInDominanceOrder(partition)The partitions of the same size that this one covers in dominance order: the upper covers of its conjugate.

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

0 if Length(_x) < 1 else Apply(
                                (u: list<integer>) => Sum(Map(
                                                               Function(Count(Filter(
                                                                                      Range(1, Length(u)),
                                                                                      Function(j > i &&
                                                                                               (i == 1 ||
                                                                                                u[i - 1] >
                                                                                                u[i]) &&
                                                                                               (j == Length(u) ||
                                                                                                u[j] >
                                                                                                u[j + 1]) &&
                                                                                               (j ==
                                                                                                i + 1 ||
                                                                                                u[i] == u[j]), j),
                                                                                    )), i),
                                                               Range(
                                                                      1,
                                                                      Length(u),
                                                                    ),
                                                             )),
                                Map(
                                     Function(
                                               Count(Filter(
                                                             Range(
                                                                    1,
                                                                    Length(_x),
                                                                  ),
                                                             Function(
                                                                       _x[j] >=
                                                                       i,
                                                                       j,
                                                                     ),
                                                           )),
                                               i,
                                             ),
                                     Range(1, Max(_x)),
                                   ),
                              )

Examples

Basic #

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

live ↗

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

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

live ↗

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

Properties #

FindStat's value for [1]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerPartition}}([1]), \text{LowerCoverCountInDominanceOrder})$Out$0$

Known value · FindStat St000480

FindStat's value for [2]

live ↗

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

Known value · FindStat St000480

FindStat's value for [3]

live ↗

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

Known value · FindStat St000480

About

Examples	5
Crosswalk	FindStat