TransitiveMonotoneFactorizationCount

Transitive monotone factorizations of genus zero of a permutation of this cycle type, into n + length - 2 transpositions.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001938via IntegerPartition✓↗

TransitiveMonotoneFactorizationCount(partition)The transitive monotone factorizations of genus zero of a permutation of this cycle type, into n + length - 2 transpositions: the product of the central binomials and of the parts, times the rising factorial of 2n + 1 of length (length - 3), or its reciprocal.

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 Product(Map(Function(Product(Range(
                                                             _x[i] + 1,
                                                             2 * _x[i],
                                                           )) /
                                              (1 if _x[i] == 0 else Product(Range(1, _x[i]))), i), Range(1, Length(_x)))) *
                         Product(_x) *
                         (1 if Length(_x) < 4 else Product(Range(
                                                                  2 * Sum(_x) +
                                                                  1,
                                                                  2 * Sum(_x) +
                                                                  Length(_x) - 3,
                                                                ))) /
                         ((2 * Sum(_x) - 1) * 2 * Sum(_x) if Length(_x) == 1 else 2 * Sum(_x) if Length(_x) == 2 else 1)

Examples

Basic #

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

live ↗

In$\mathrm{TransitiveMonotoneFactorizationCount}({\operatorname{IntegerPartition}}([4, 2, 1]))$Out$6\,720$

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

live ↗

In$\mathrm{TransitiveMonotoneFactorizationCount}({\operatorname{IntegerPartition}}([3, 3, 1]))$Out$7\,200$

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St001938

FindStat's value for [2]

live ↗

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

Known value · FindStat St001938

FindStat's value for [3]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerPartition}}([3]), \text{TransitiveMonotoneFactorizationCount})$Out$2$

Known value · FindStat St001938

About

Examples	5
Crosswalk	FindStat