NonDominatingSubmultisetCount

The distinct sub-multisets of the parts that, read as partitions, do not dominate their own conjugate.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001714via IntegerPartition✓↗

NonDominatingSubmultisetCount(partition)The distinct sub-multisets of the parts that, read as partitions, do not dominate their own 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.
• FindStat calls the sub-multisets subpartitions. The partitions with none are the wide ones.

Definition

Epsil, as the implementation runs it.

on IntegerPartition, over _x

0 if Length(_x) < 1 else Apply(
                                (m: list<integer>) => Apply(
                                                             (r: list<integer>) => Apply(
                                                                                          (s: list<integer>) => Max(Sum(Map(
                                                                                                                             Function(Min(
                                                                                                                                           1,
                                                                                                                                           Max(Sum(Map(Function(1 if Max(Sum(Map(Function((1 if r[i] <=
                                                                                                                                                                                                (t - 1 -
                                                                                                                                                                                                 (t - 1) %
                                                                                                                                                                                                 s[i]) /
                                                                                                                                                                                                s[i] %
                                                                                                                                                                                                (m[i] + 1) else 0) *
                                                                                                                                                                                          _x[i] *
                                                                                                                                                                                          (1 if Max(Sum(Map(Function(1 if r[j] <=
                                                                                                                                                                                                                          (t - 1 -
                                                                                                                                                                                                                           (t - 1) %
                                                                                                                                                                                                                           s[j]) /
                                                                                                                                                                                                                          s[j] %
                                                                                                                                                                                                                          (m[j] + 1) else 0, j), Range(1, i))), 0) <=
                                                                                                                                                                                                k else 0), i), Range(1, Length(_x)))), 0) <
                                                                                                                                                                     Max(Sum(Map(Function((1 if r[i] <=
                                                                                                                                                                                                (t - 1 -
                                                                                                                                                                                                 (t - 1) %
                                                                                                                                                                                                 s[i]) /
                                                                                                                                                                                                s[i] %
                                                                                                                                                                                                (m[i] + 1) else 0) *
                                                                                                                                                                                          Min(_x[i], k), i), Range(1, Length(_x)))), 0) else 0, k), Range(1, Sum(_x)))), 0),
                                                                                                                                         ), t),
                                                                                                                             Range(1, Product(Map(Function(m[j] + 1 if r[j] == m[j] else 1, j), Range(1, Length(_x))))),
                                                                                                                           )), 0),
                                                                                          Map(
                                                                                               Function(Product(Map(
                                                                                                                     Function(m[j] + 1 if _x[j] > _x[i] &&
                                                                                                                                          r[j] == m[j] else 1, j),
                                                                                                                     Range(1, Length(_x)),
                                                                                                                   )), i),
                                                                                               Range(1, Length(_x)),
                                                                                             ),
                                                                                        ),
                                                             Map(
                                                                  Function(Count(Filter(
                                                                                         Range(1, i),
                                                                                         Function(_x[j] == _x[i], j),
                                                                                       )), i),
                                                                  Range(
                                                                         1,
                                                                         Length(_x),
                                                                       ),
                                                                ),
                                                           ),
                                Map(
                                     Function(
                                               Count(Filter(
                                                             Range(
                                                                    1,
                                                                    Length(_x),
                                                                  ),
                                                             Function(_x[j] ==
                                                                      _x[i], j),
                                                           )),
                                               i,
                                             ),
                                     Range(1, Length(_x)),
                                   ),
                              )

Examples

Basic #

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

live ↗

In$\mathrm{NonDominatingSubmultisetCount}({\operatorname{IntegerPartition}}([4, 2, 1]))$Out$0$

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

live ↗

In$\mathrm{NonDominatingSubmultisetCount}({\operatorname{IntegerPartition}}([3, 3, 1]))$Out$0$

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St001714

FindStat's value for [2]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerPartition}}([2]), \text{NonDominatingSubmultisetCount})$Out$0$

Known value · FindStat St001714

FindStat's value for [3]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerPartition}}([3]), \text{NonDominatingSubmultisetCount})$Out$0$

Known value · FindStat St001714

About

Examples	5
Crosswalk	FindStat