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- 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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| Original file line number | Diff line number | Diff line change |
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| @@ -1,4 +1,4 @@ | ||
| julia 0.4 | ||
| julia 0.5- | ||
| Compat | ||
| Polynomials | ||
| Iterators |
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| export combinations | ||
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| #The Combinations iterator | ||
| import Base: start, next, done, length | ||
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| immutable Combinations{T} | ||
| a::T | ||
| t::Int | ||
| end | ||
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| start(c::Combinations) = [1:c.t;] | ||
| function next(c::Combinations, s) | ||
| comb = [c.a[si] for si in s] | ||
| if c.t == 0 | ||
| # special case to generate 1 result for t==0 | ||
| return (comb,[length(c.a)+2]) | ||
| end | ||
| s = copy(s) | ||
| for i = length(s):-1:1 | ||
| s[i] += 1 | ||
| if s[i] > (length(c.a) - (length(s)-i)) | ||
| continue | ||
| end | ||
| for j = i+1:endof(s) | ||
| s[j] = s[j-1]+1 | ||
| end | ||
| break | ||
| end | ||
| (comb,s) | ||
| end | ||
| 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) | ||
| if t < 0 | ||
| # generate 0 combinations for negative argument | ||
| t = length(a)+1 | ||
| end | ||
| Combinations(a, t) | ||
| end | ||
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| #generate combinations of all orders, chaining of order iterators is eager, | ||
| #but sequence at each order is lazy | ||
| combinations(a) = chain([combinations(a,k) for k=1:length(a)]...) | ||
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| #Factorials and elementary coefficients | ||
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| export | ||
| derangement, | ||
| subfactorial, | ||
| doublefactorial, | ||
| hyperfactorial, | ||
| multifactorial, | ||
| gamma, | ||
| primorial, | ||
| multinomial | ||
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| # The number of permutations of n with no fixed points (subfactorial) | ||
| function derangement(sn::Integer) | ||
| n = BigInt(sn) | ||
| return num(factorial(n)*sum([(-1)^k//factorial(k) for k=0:n])) | ||
| end | ||
| subfactorial(n::Integer) = derangement(n) | ||
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| function doublefactorial(n::Integer) | ||
| if n < 0 | ||
| throw(DomainError()) | ||
| end | ||
| z = BigInt() | ||
| ccall((:__gmpz_2fac_ui, :libgmp), Void, | ||
| (Ptr{BigInt}, UInt), &z, @compat(UInt(n))) | ||
| return z | ||
| end | ||
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| # Hyperfactorial | ||
| hyperfactorial(n::Integer) = prod([i^i for i = BigInt(2):n]) | ||
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| function multifactorial(n::Integer, m::Integer) | ||
| if n < 0 | ||
| throw(DomainError()) | ||
| end | ||
| z = BigInt() | ||
| ccall((:__gmpz_mfac_uiui, :libgmp), Void, | ||
| (Ptr{BigInt}, UInt, UInt), &z, @compat(UInt(n)), @compat(UInt(m))) | ||
| return z | ||
| end | ||
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| function primorial(n::Integer) | ||
| if n < 0 | ||
| throw(DomainError()) | ||
| end | ||
| z = BigInt() | ||
| ccall((:__gmpz_primorial_ui, :libgmp), Void, | ||
| (Ptr{BigInt}, UInt), &z, @compat(UInt(n))) | ||
| return z | ||
| end | ||
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| #Multinomial coefficient where n = sum(k) | ||
| function multinomial(k...) | ||
| s = 0 | ||
| result = 1 | ||
| for i in k | ||
| s += i | ||
| result *= binomial(s, i) | ||
| end | ||
| result | ||
| end | ||
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| #Special named numbers and symbols | ||
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| export bellnum, | ||
| catalannum, | ||
| fibonaccinum, | ||
| jacobisymbol, | ||
| lassallenum, | ||
| legendresymbol, | ||
| lucasnum, | ||
| stirlings1 | ||
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| # Returns the n-th Bell number | ||
| function bellnum(bn::Integer) | ||
| if bn < 0 | ||
| throw(DomainError()) | ||
| else | ||
| n = BigInt(bn) | ||
| end | ||
| list = Array(BigInt, div(n*(n+1), 2)) | ||
| list[1] = 1 | ||
| for i = 2:n | ||
| beg = div(i*(i-1),2) | ||
| list[beg+1] = list[beg] | ||
| for j = 2:i | ||
| list[beg+j] = list[beg+j-1]+list[beg+j-i] | ||
| end | ||
| end | ||
| return list[end] | ||
| end | ||
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| # Returns the n-th Catalan number | ||
| function catalannum(bn::Integer) | ||
| if bn<0 | ||
| throw(DomainError()) | ||
| else | ||
| n = BigInt(bn) | ||
| end | ||
| div(binomial(2*n, n), (n + 1)) | ||
| end | ||
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| function fibonaccinum(n::Integer) | ||
| if n < 0 | ||
| throw(DomainError()) | ||
| end | ||
| z = BigInt() | ||
| ccall((:__gmpz_fib_ui, :libgmp), Void, | ||
| (Ptr{BigInt}, UInt), &z, @compat(UInt(n))) | ||
| return z | ||
| end | ||
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| function jacobisymbol(a::Integer, b::Integer) | ||
| ba = BigInt(a) | ||
| bb = BigInt(b) | ||
| return ccall((:__gmpz_jacobi, :libgmp), Cint, | ||
| (Ptr{BigInt}, Ptr{BigInt}), &ba, &bb) | ||
| end | ||
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| #Computes Lassalle's sequence | ||
| #OEIS entry A180874 | ||
| function lassallenum(m::Integer) | ||
| A = ones(BigInt,m) | ||
| for n=2:m | ||
| 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])) | ||
| end | ||
| A[m] | ||
| end | ||
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| function legendresymbol(a::Integer, b::Integer) | ||
| ba = BigInt(a) | ||
| bb = BigInt(b) | ||
| return ccall((:__gmpz_legendre, :libgmp), Cint, | ||
| (Ptr{BigInt}, Ptr{BigInt}), &ba, &bb) | ||
| end | ||
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| function lucasnum(n::Integer) | ||
| if n < 0 | ||
| throw(DomainError()) | ||
| end | ||
| z = BigInt() | ||
| ccall((:__gmpz_lucnum_ui, :libgmp), Void, | ||
| (Ptr{BigInt}, UInt), &z, @compat(UInt(n))) | ||
| return z | ||
| end | ||
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| # Returns s(n, k), the signed Stirling number of first kind | ||
| function stirlings1(n::Integer, k::Integer) | ||
| p = poly(0:(n-1)) | ||
| p[n - k + 1] | ||
| end | ||
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