UnsplittableFactorCount

The number of unsplittable factors of the set partition under the split product.

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UnsplittableFactorCount(partition)The number of unsplittable factors of the set partition under the split product.

Typed signatures

Type	Package	Applies
(set_partition) -> number	combinatorics	always

Domain: Set partition statistics

Details

• Defined over SetPartition as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
• Takes a SetPartition 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's St000823. The split product of set partitions of m and n glues the i-th block of the first to the i-th block of the second, shifted by m.

Definition

Epsil, as the implementation runs it.

on SetPartition, over _x

0 if Length(_x) < 1 else 1 if Length(Flatten(_x)) < 2 else 1 + Sum(Map(
                                                                        Function(1 if Length(_x) < 2 else 1 if Max(Sum(Map(Function(1 if (_x[k][Max(Sum(Map(Function(1 if _x[k][j] <= m else 0, j), Range(1, Length(_x[k])))), 0) +
                                                                                                                                                1] if Last(_x[k]) > m else Length(Flatten(_x)) + 1) >
                                                                                                                                         (_x[k + 1][Max(Sum(Map(
                                                                                                                                                                 Function(1 if _x[k + 1][j] <=
                                                                                                                                                                               m else 0, j),
                                                                                                                                                                 Range(1, Length(_x[k + 1])),
                                                                                                                                                               )), 0) +
                                                                                                                                                    1] if Last(_x[k + 1]) >
                                                                                                                                                         m else Length(Flatten(_x)) + 1) else 0, k), Range(1, Length(_x) - 1, 1))), 0) ==
                                                                                                               0 else 0, m),
                                                                        Range(
                                                                               1,
                                                                               Length(Flatten(_x)) -
                                                                               1,
                                                                               1,
                                                                             ),
                                                                      ))

Examples

Basic #

the partition $\{1,3\} \mid \{2\}$

live ↗

In$\mathrm{UnsplittableFactorCount}({\operatorname{SetPartition}}([[1, 3], [2]]))$Out$2$

the partition $\{1\} \mid \{2,4\} \mid \{3\}$

live ↗

In$\mathrm{UnsplittableFactorCount}({\operatorname{SetPartition}}([[1], [2, 4], [3]]))$Out$1$

Properties #

FindStat's value for {{1,2}}

live ↗

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

Known value · FindStat St000823

FindStat's value for {{1,2,3}}

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{SetPartition}}([[1, 2, 3]]), \text{UnsplittableFactorCount})$Out$3$

Known value · FindStat St000823

FindStat's value for {{1},{2}}

live ↗

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

Known value · FindStat St000823

About

Examples	5
Crosswalk	FindStat