StarTranspositionFactorizationCount

Minimal transitive factorizations of a permutation of this cycle type into star transpositions: the product of the parts times (n + 1) ... (n + length - 2).

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001936via IntegerPartition✓↗

StarTranspositionFactorizationCount(partition)The minimal transitive factorizations of a permutation of this cycle type into star transpositions: the product of the parts times (n + 1)...(n + length - 2).

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 1 if Length(_x) == 1 else Product(_x) * (1 if Length(_x) ==
                                                                       2 else Product(Range(
                                                                                            Sum(_x) + 1,
                                                                                            Sum(_x) + Length(_x) -
                                                                                            2,
                                                                                          )))

Examples

Basic #

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

live ↗

In$\mathrm{StarTranspositionFactorizationCount}({\operatorname{IntegerPartition}}([4, 2, 1]))$Out$64$

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

live ↗

In$\mathrm{StarTranspositionFactorizationCount}({\operatorname{IntegerPartition}}([3, 3, 1]))$Out$72$

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St001936

FindStat's value for [2]

live ↗

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

Known value · FindStat St001936

FindStat's value for [3]

live ↗

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

Known value · FindStat St001936

About

Examples	5
Crosswalk	FindStat