Combinatorial Functions

EqualitySampler.stirlings2 — Function

stirlings2(n::Integer, k::Integer) -> Any

Compute the Stirlings numbers of the second kind. The EqualitySampler.ExplicitStrategy (default) uses an explicit loop and is computationally more efficient but subject to overflow, so using BigInt is advised. The EqualitySampler.RecursiveStrategy uses recursion and is mathematically elegant yet inefficient for large values.

EqualitySampler.logstirlings2 — Function

logstirlings2(n::Integer, k::Integer) -> Any

Compute the logarithm of the Stirlings numbers of the second kind with an explicit formula.

EqualitySampler.stirlings2r — Function

stirlings2r(n::T, k::T, r::T) where T <: Integer
stirlings2r(n::T, k::T, r::T, ::Type{EqualitySampler.ExplicitStrategy})  where T <: Integer
stirlings2r(n::T, k::T, r::T, ::Type{EqualitySampler.RecursiveStrategy}) where T <: Integer

Compute the r-Stirlings numbers of the second kind. The EqualitySampler.ExplicitStrategy (default) uses an explicit loop and is computationally more efficient but subject to overflow, so using BigInt is advised. The EqualitySampler.RecursiveStrategy uses recursion and is mathematically elegant yet inefficient for large values.

EqualitySampler.logstirlings2r — Function

logstirlings2r(n::T, k::T, r::T) where T <: Integer

Computes the logarithm of the r-stirling numbers with an explicit formula.

EqualitySampler.unsignedstirlings1 — Function

unsignedstirlings1(n::Integer, k::Integer) -> Any

Compute the absolute value of the Stirlings numbers of the first kind. The EqualitySampler.ExplicitStrategy (default) uses an explicit loop and is computationally more efficient but subject to overflow, so using BigInt is advised. The EqualitySampler.RecursiveStrategy uses recursion and is mathematically elegant yet inefficient for large values.

EqualitySampler.logunsignedstirlings1 — Function

logunsignedstirlings1(n::Integer, k::Integer) -> Any

Compute the logarithm of the absolute value of the Stirlings numbers of the first kind.

EqualitySampler.bellnum — Function

bellnum(n::Integer) -> Any

Computes the Bell numbers.

EqualitySampler.bellnumr — Function

bellnumr(n::Integer, r::Integer) -> Any

Computes the $r$-Bell numbers.

EqualitySampler.logbellnumr — Function

logbellnumr(n::Integer, r::Integer) -> Any

Computes the logarithm of the $r$-Bell numbers.

EqualitySampler.PartitionSpace — Type

struct PartitionSpace{T<:Integer, P<:EqualitySampler.AbstractPartitionSpace}

# constructor
PartitionSpace(k::T, ::Type{U} = EqualitySampler.DistinctPartitionSpace)

A type to represent the space of partitions. EqualitySampler.AbstractPartitionSpace indicates whether partitions should contains duplicates or be distinct. For example, the distinct iterator will return [1, 1, 2] but not [2, 2, 1] and [1, 1, 3], which are returned when P = EqualitySampler.DuplicatedPartitionSpace.