LagrangeInversionCoefficient

The coefficient of the monomial in the compositional-inverse series: n (n + 1) ... (n + length - 2) over the factorials of the multiplicities.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001967via IntegerPartition✓↗

LagrangeInversionCoefficient(partition)The coefficient of the monomial in the compositional-inverse series: the product of the size plus 0 to the number of parts less 2, over the product of the factorials of the multiplicities.

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(Map(
                                                                 Function(Sum(_x) +
                                                                          k, k),
                                                                 Range(
                                                                        0,
                                                                        Length(_x) -
                                                                        2,
                                                                      ),
                                                               ))) / Product(Map(
                                                                                 Function(Count(Filter(
                                                                                                        Range(1, i),
                                                                                                        Function(_x[j] == _x[i], j),
                                                                                                      )), i),
                                                                                 Range(1, Length(_x)),
                                                                               ))

Examples

Basic #

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

live ↗

In$\mathrm{LagrangeInversionCoefficient}({\operatorname{IntegerPartition}}([4, 2, 1]))$Out$56$

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

live ↗

In$\mathrm{LagrangeInversionCoefficient}({\operatorname{IntegerPartition}}([3, 3, 1]))$Out$28$

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St001967

FindStat's value for [2]

live ↗

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

Known value · FindStat St001967

FindStat's value for [3]

live ↗

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

Known value · FindStat St001967

About

Examples	5
Crosswalk	FindStat