LogarithmicDerivativeCoefficient

The coefficient of the monomial in the logarithmic derivative of 1 - c_1 x - c_2 x^2/2! - ..., times a factorial: the set partitions with these block sizes, times (length - 1)! for arranging the blocks in a cycle.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001968via IntegerPartition✓↗

LogarithmicDerivativeCoefficient(partition)The coefficient of the monomial in the logarithmic derivative of 1 - c_1 x - c_2 x^2/2! - ..., times the factorial: the multinomial of the parts times the rearrangements of the parts, over the number of parts.

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 Sum(_x) == 0 else Product(Range(1, Sum(_x)))) /
                         Product(Map(Function(
                                               1 if _x[i] == 0 else Product(Range(1, _x[i])),
                                               i,
                                             ), Range(1, Length(_x)))) * ((1 if Length(_x) == 0 else Product(Range(
                                                                       1,
                                                                       Length(_x),
                                                                     ))) /
                             Product(Map(
                                          Function(Count(Filter(
                                                                 Range(1, i),
                                                                 Function(_x[j] ==
                                                                          _x[i], j),
                                                               )), i),
                                          Range(1, Length(_x)),
                                        ))) / Length(_x)

Examples

Basic #

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

live ↗

In$\mathrm{LogarithmicDerivativeCoefficient}({\operatorname{IntegerPartition}}([4, 2, 1]))$Out$210$

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

live ↗

In$\mathrm{LogarithmicDerivativeCoefficient}({\operatorname{IntegerPartition}}([3, 3, 1]))$Out$140$

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St001968

FindStat's value for [2]

live ↗

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

Known value · FindStat St001968

FindStat's value for [3]

live ↗

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

Known value · FindStat St001968

About

Examples	5
Crosswalk	FindStat