MonotoneHurwitzNumber

The monotone factorizations of genus zero of a permutation of this cycle type into transpositions: the product of the Catalan numbers C(p - 1) over the parts p.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001934via IntegerPartition✓↗

MonotoneHurwitzNumber(partition)The monotone factorizations of genus zero of a permutation of this cycle type into transpositions: the product of the Catalan numbers C(p - 1) over the parts p.

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.
• The cycles factor independently, and the larger entries of the transpositions, weakly increasing, fix how the factors interleave.

Definition

Epsil, as the implementation runs it.

on IntegerPartition, over _x

1 if Length(_x) < 1 else Apply(
                                (c: list<integer>) => Product(Map(
                                                                   Function(
                                                                             c[_x[i]],
                                                                             i,
                                                                           ),
                                                                   Range(
                                                                          1,
                                                                          Length(_x),
                                                                        ),
                                                                 )),
                                Fold(Function(Join(
                                                    a,
                                                    [Sum(Map(
                                                              Function(a[i] *
                                                                       a[Length(a) -
                                                                         (i - 1)], i),
                                                              Range(
                                                                     1,
                                                                     Length(a),
                                                                   ),
                                                            ))],
                                                  ), a, b), [1], Range(
                                                                       1,
                                                                       Max(_x),
                                                                     )),
                              )

Examples

Basic #

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

live ↗

In$\mathrm{MonotoneHurwitzNumber}({\operatorname{IntegerPartition}}([4, 2, 1]))$Out$5$

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

live ↗

In$\mathrm{MonotoneHurwitzNumber}({\operatorname{IntegerPartition}}([3, 3, 1]))$Out$4$

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St001934

FindStat's value for [2]

live ↗

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

Known value · FindStat St001934

FindStat's value for [3]

live ↗

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

Known value · FindStat St001934

About

Examples	5
Crosswalk	FindStat