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Major code reorganization
- DROP 0.4 SUPPORT - Import most of base/combinatorics.jl (Ref: JuliaLang/julia#13897) - Move most of the special numbers to numbers.jl - Put combinations, permutations and partitions in their own files - Rename special numbers with ~num suffix. This renaming is particularly important for catalannum to avoid clashing with the Base.catalan irrational constant.
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‎README.md

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# Combinatorics
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[![Combinatorics](http://pkg.julialang.org/badges/Combinatorics_0.3.svg)](http://pkg.julialang.org/?pkg=Combinatorics&ver=0.3)
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[![Combinatorics](http://pkg.julialang.org/badges/Combinatorics_0.4.svg)](http://pkg.julialang.org/?pkg=Combinatorics&ver=0.4)
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[![Build Status](https://travis-ci.org/JuliaLang/Combinatorics.jl.svg?branch=master)](https://travis-ci.org/JuliaLang/Combinatorics.jl)
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[![Coverage Status](https://coveralls.io/repos/JuliaLang/Combinatorics.jl/badge.svg?branch=master&service=github)](https://coveralls.io/github/JuliaLang/Combinatorics.jl?branch=master)

‎REQUIRE

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julia 0.4
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julia 0.5-
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Compat
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Polynomials
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Iterators

‎src/Combinatorics.jl

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using Compat, Polynomials, Iterators
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import Base:combinations
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export bell,
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derangement,
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doublefactorial,
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fibonacci,
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hyperfactorial,
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jacobisymbol,
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lassalle,
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legendresymbol,
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lucas,
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multifactorial,
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multinomial,
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primorial,
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stirlings1,
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subfactorial
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include("numbers.jl")
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include("factorials.jl")
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include("combinations.jl")
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include("permutations.jl")
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include("partitions.jl")
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include("youngdiagrams.jl")
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# Returns the n-th Bell number
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function bell(bn::Integer)
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if bn < 0
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throw(DomainError())
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else
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n = BigInt(bn)
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end
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list = Array(BigInt, div(n*(n+1), 2))
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list[1] = 1
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for i = 2:n
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beg = div(i*(i-1),2)
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list[beg+1] = list[beg]
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for j = 2:i
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list[beg+j] = list[beg+j-1]+list[beg+j-i]
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end
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end
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return list[end]
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end
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# Returns the n-th Catalan number
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function catalan(bn::Integer)
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if bn<0
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throw(DomainError())
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else
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n = BigInt(bn)
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end
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div(binomial(2*n, n), (n + 1))
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end
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#generate combinations of all orders, chaining of order iterators is eager,
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#but sequence at each order is lazy
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combinations(a) = chain([combinations(a,k) for k=1:length(a)]...)
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# The number of permutations of n with no fixed points (subfactorial)
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function derangement(sn::Integer)
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n = BigInt(sn)
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return num(factorial(n)*sum([(-1)^k//factorial(k) for k=0:n]))
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end
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subfactorial(n::Integer) = derangement(n)
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function doublefactorial(n::Integer)
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if n < 0
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throw(DomainError())
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end
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z = BigInt()
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ccall((:__gmpz_2fac_ui, :libgmp), Void,
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(Ptr{BigInt}, UInt), &z, @compat(UInt(n)))
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return z
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end
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function fibonacci(n::Integer)
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if n < 0
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throw(DomainError())
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end
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z = BigInt()
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ccall((:__gmpz_fib_ui, :libgmp), Void,
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(Ptr{BigInt}, UInt), &z, @compat(UInt(n)))
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return z
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end
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# Hyperfactorial
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hyperfactorial(n::Integer) = prod([i^i for i = BigInt(2):n])
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function jacobisymbol(a::Integer, b::Integer)
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ba = BigInt(a)
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bb = BigInt(b)
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return ccall((:__gmpz_jacobi, :libgmp), Cint,
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(Ptr{BigInt}, Ptr{BigInt}), &ba, &bb)
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end
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#Computes Lassalle's sequence
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#OEIS entry A180874
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function lassalle(m::Integer)
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A = ones(BigInt,m)
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for n=2:m
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A[n]=(-1)^(n-1) * (catalan(n) + sum([(-1)^j*binomial(2n-1, 2j-1)*A[j]*catalan(n-j) for j=1:n-1]))
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end
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A[m]
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end
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function legendresymbol(a::Integer, b::Integer)
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ba = BigInt(a)
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bb = BigInt(b)
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return ccall((:__gmpz_legendre, :libgmp), Cint,
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(Ptr{BigInt}, Ptr{BigInt}), &ba, &bb)
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end
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function lucas(n::Integer)
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if n < 0
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throw(DomainError())
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end
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z = BigInt()
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ccall((:__gmpz_lucnum_ui, :libgmp), Void,
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(Ptr{BigInt}, UInt), &z, @compat(UInt(n)))
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return z
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end
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function multifactorial(n::Integer, m::Integer)
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if n < 0
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throw(DomainError())
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end
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z = BigInt()
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ccall((:__gmpz_mfac_uiui, :libgmp), Void,
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(Ptr{BigInt}, UInt, UInt), &z, @compat(UInt(n)), @compat(UInt(m)))
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return z
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end
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# Multinomial coefficient where n = sum(k)
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function multinomial(k...)
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s = 0
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result = 1
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for i in k
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s += i
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result *= binomial(s, i)
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end
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result
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end
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function primorial(n::Integer)
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if n < 0
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throw(DomainError())
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end
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z = BigInt()
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ccall((:__gmpz_primorial_ui, :libgmp), Void,
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(Ptr{BigInt}, UInt), &z, @compat(UInt(n)))
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return z
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end
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# Returns s(n, k), the signed Stirling number of first kind
154-
function stirlings1(n::Integer, k::Integer)
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p = poly(0:(n-1))
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p[n - k + 1]
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end
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end # module
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end #module

‎src/combinations.jl

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export combinations
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#The Combinations iterator
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import Base: start, next, done, length
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immutable Combinations{T}
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a::T
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t::Int
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end
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start(c::Combinations) = [1:c.t;]
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function next(c::Combinations, s)
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comb = [c.a[si] for si in s]
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if c.t == 0
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# special case to generate 1 result for t==0
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return (comb,[length(c.a)+2])
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end
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s = copy(s)
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for i = length(s):-1:1
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s[i] += 1
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if s[i] > (length(c.a) - (length(s)-i))
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continue
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end
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for j = i+1:endof(s)
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s[j] = s[j-1]+1
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end
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break
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end
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(comb,s)
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end
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done(c::Combinations, s) = !isempty(s) && s[1] > length(c.a)-c.t+1
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length(c::Combinations) = binomial(length(c.a),c.t)
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eltype{T}(::Type{Combinations{T}}) = Vector{eltype(T)}
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function combinations(a, t::Integer)
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if t < 0
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# generate 0 combinations for negative argument
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t = length(a)+1
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end
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Combinations(a, t)
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end
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#generate combinations of all orders, chaining of order iterators is eager,
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#but sequence at each order is lazy
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combinations(a) = chain([combinations(a,k) for k=1:length(a)]...)
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‎src/factorials.jl

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#Factorials and elementary coefficients
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export
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derangement,
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subfactorial,
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doublefactorial,
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hyperfactorial,
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multifactorial,
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gamma,
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primorial,
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multinomial
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# The number of permutations of n with no fixed points (subfactorial)
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function derangement(sn::Integer)
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n = BigInt(sn)
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return num(factorial(n)*sum([(-1)^k//factorial(k) for k=0:n]))
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end
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subfactorial(n::Integer) = derangement(n)
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function doublefactorial(n::Integer)
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if n < 0
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throw(DomainError())
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end
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z = BigInt()
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ccall((:__gmpz_2fac_ui, :libgmp), Void,
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(Ptr{BigInt}, UInt), &z, @compat(UInt(n)))
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return z
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end
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# Hyperfactorial
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hyperfactorial(n::Integer) = prod([i^i for i = BigInt(2):n])
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function multifactorial(n::Integer, m::Integer)
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if n < 0
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throw(DomainError())
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end
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z = BigInt()
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ccall((:__gmpz_mfac_uiui, :libgmp), Void,
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(Ptr{BigInt}, UInt, UInt), &z, @compat(UInt(n)), @compat(UInt(m)))
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return z
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end
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function primorial(n::Integer)
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if n < 0
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throw(DomainError())
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end
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z = BigInt()
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ccall((:__gmpz_primorial_ui, :libgmp), Void,
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(Ptr{BigInt}, UInt), &z, @compat(UInt(n)))
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return z
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end
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#Multinomial coefficient where n = sum(k)
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function multinomial(k...)
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s = 0
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result = 1
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for i in k
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s += i
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result *= binomial(s, i)
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end
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result
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end
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‎src/numbers.jl

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#Special named numbers and symbols
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export bellnum,
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catalannum,
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fibonaccinum,
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jacobisymbol,
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lassallenum,
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legendresymbol,
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lucasnum,
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stirlings1
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# Returns the n-th Bell number
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function bellnum(bn::Integer)
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if bn < 0
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throw(DomainError())
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else
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n = BigInt(bn)
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end
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list = Array(BigInt, div(n*(n+1), 2))
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list[1] = 1
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for i = 2:n
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beg = div(i*(i-1),2)
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list[beg+1] = list[beg]
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for j = 2:i
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list[beg+j] = list[beg+j-1]+list[beg+j-i]
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end
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end
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return list[end]
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end
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# Returns the n-th Catalan number
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function catalannum(bn::Integer)
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if bn<0
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throw(DomainError())
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else
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n = BigInt(bn)
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end
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div(binomial(2*n, n), (n + 1))
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end
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function fibonaccinum(n::Integer)
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if n < 0
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throw(DomainError())
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end
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z = BigInt()
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ccall((:__gmpz_fib_ui, :libgmp), Void,
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(Ptr{BigInt}, UInt), &z, @compat(UInt(n)))
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return z
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end
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function jacobisymbol(a::Integer, b::Integer)
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ba = BigInt(a)
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bb = BigInt(b)
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return ccall((:__gmpz_jacobi, :libgmp), Cint,
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(Ptr{BigInt}, Ptr{BigInt}), &ba, &bb)
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end
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#Computes Lassalle's sequence
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#OEIS entry A180874
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function lassallenum(m::Integer)
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A = ones(BigInt,m)
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for n=2:m
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A[n]=(-1)^(n-1) * (catalannum(n) + sum([(-1)^j*binomial(2n-1, 2j-1)*A[j]*catalannum(n-j) for j=1:n-1]))
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end
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A[m]
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end
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function legendresymbol(a::Integer, b::Integer)
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ba = BigInt(a)
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bb = BigInt(b)
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return ccall((:__gmpz_legendre, :libgmp), Cint,
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(Ptr{BigInt}, Ptr{BigInt}), &ba, &bb)
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end
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function lucasnum(n::Integer)
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if n < 0
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throw(DomainError())
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end
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z = BigInt()
81+
ccall((:__gmpz_lucnum_ui, :libgmp), Void,
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(Ptr{BigInt}, UInt), &z, @compat(UInt(n)))
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return z
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end
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# Returns s(n, k), the signed Stirling number of first kind
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function stirlings1(n::Integer, k::Integer)
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p = poly(0:(n-1))
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p[n - k + 1]
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end
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