Converted partition functions from tasks to iterators

* Renamed integer_partitions -> partitions (multiple dispatch!)
* Added partitions(n) variant to generate all integer partitions
* Added npartitions, nsetpartitions functions
* Updated docs

base/combinatorics.jl

@@ -252,110 +252,202 @@ end
 done(p::Permutations, s) = !isempty(s) && s[1] > length(p.a)
 
 
-# Algorithm H from TAoCP 7.2.1.4
-# Partition n into m parts
-function integer_partitions{T<:Integer}(n::T, m::T)
-  if n < m || m < 2
-    throw("Require n >= m >= 2!")
-  end
-  # H1
-  a = [n - m + 1, ones(T, m), -1]
-  # H2
-  while true
-    produce(a[1:m])
-    if a[2] < a[1] - 1
-      # H3
-      a[1] = a[1] - 1
-      a[2] = a[2] + 1
-      continue # to H2
+# Integer Partitions
+
+immutable IntegerPartitions
+    n::Int
+end
+
+length(p::IntegerPartitions) = npartitions(p.n)
+
+partitions(n::Integer) = IntegerPartitions(n)
+
+start(p::IntegerPartitions) = Int[]
+done(p::IntegerPartitions, xs) = length(xs) == p.n
+next(p::IntegerPartitions, xs) = (xs = nextpartition(p.n,xs); (xs,xs))
+
+function nextpartition(n, as)
+    if isempty(as);  return Int[n];  end
+
+    xs = similar(as,0)
+    sizehint(xs,length(as)+1)
+
+    for i = 1:length(as)-1
+        if as[i+1] == 1
+            x = as[i]-1
+            push!(xs, x)
+            n -= x
+            while n > x
+                push!(xs, x)
+                n -= x
+            end
+            push!(xs, n)
+
+            return xs
+        end
+        push!(xs, as[i])
+        n -= as[i]
     end
-    # H4
-    j = 3
-    s = a[1] + a[2] - 1
-    if a[j] >= a[1] - 1
-      while true
-        s = s + a[j]
-        j = j + 1
-        if a[j] < a[1] - 1
-          break # end loop
+    push!(xs, as[end]-1)
+    push!(xs, 1)
+
+    xs
+end
+
+const _npartitions = (Int=>Int)[]
+function npartitions(n::Int)
+    if n < 0
+        0
+    elseif n < 2
+        1
+    elseif (np = get(_npartitions, n, 0)) > 0
+        np
+    else
+        np = 0
+        sgn = 1
+        for k = 1:n
+            np += sgn * (npartitions(n-k*(3k-1)>>1) + npartitions(n-k*(3k+1)>>1))
+            sgn = -sgn
         end
-      end
+        _npartitions[n] = np
     end
-    # H5
-    if j > m 
-      break # terminate
+end
+
+
+# Algorithm H from TAoCP 7.2.1.4
+# Partition n into m parts
+# in colex order (lexicographic by reflected sequence)
+
+immutable FixedPartitions
+    n::Int
+    m::Int
+end
+
+length(f::FixedPartitions) = npartitions(f.n,f.m)
+
+partitions(n::Integer, m::Integer) = (@assert 2 <= m <= n; FixedPartitions(n,m))
+
+start(f::FixedPartitions) = Int[]
+done(f::FixedPartitions, s::Vector{Int}) = !isempty(s) && s[1]-1 <= s[end]
+next(f::FixedPartitions, s::Vector{Int}) = (xs = nextfixedpartition(f.n,f.m,s); (xs,xs))
+
+function nextfixedpartition(n, m, bs)
+    as = copy(bs)
+    if isempty(as)
+        # First iteration
+        as = [n-m+1, ones(Int, m-1)]
+    elseif as[2] < as[1]-1
+        # Most common iteration
+        as[1] -= 1
+        as[2] += 1
+    else
+        # Iterate
+        local j
+        s = as[1]+as[2]-1
+        for j = 3:m
+            if as[j] < as[1]-1; break; end
+            s += as[j]
+        end
+        x = as[j] += 1
+        for k = j-1:-1:2
+            as[k] = x
+            s -= x
+        end
+        as[1] = s
     end
-    x = a[j] + 1
-    a[j] = x
-    j = j - 1
-    # H6
-    while j > 1
-      a[j] = x
-      s = s - x
-      j = j - 1
+
+    return as
+end
+
+const _nipartitions = ((Int,Int)=>Int)[]
+function npartitions(n::Int,m::Int)
+    if n < m || m == 0
+        0
+    elseif n == m
+        1
+    elseif (np = get(_nipartitions, (n,m), 0)) > 0
+        np
+    else
+        _nipartitions[(n,m)] = npartitions(n-1,m-1) + npartitions(n-m,m)
     end
-    a[1] = s
-  end
 end
 
+
 # Algorithm H from TAoCP 7.2.1.5
 # Set partitions
-function partitions{T}(s::AbstractVector{T})
-  n = length(s)
-  n == 0 && return
-  if n == 1
-    produce(Array{T,1}[s])
-    return
-  end
-
-  # H1
-  a = zeros(Int,n)
-  b = ones(Int,n-1)
-  m = 1
-
-  while true
-    # H2
-    # convert from restricted growth string a[1:n] to set of sets
-    temp = [ Array(T,0) for k = 1:n ]
-    for k = 1:n
-      push!(temp[a[k]+1], s[k])
-    end
-    result = Array(Array{T,1},0)
-    for arr in temp
-      if !isempty(arr)
-        push!(result, arr)
-      end
-    end
-    #produce(a[1:n]) # this is the string representing set assignment
-    produce(result)
 
-    if a[n] != m
-      # H3
-      a[n] = a[n] + 1
-      continue # to H2
-    end
-    # H4
-    j = n - 1
-    while a[j] == b[j]
-      j = j - 1
+immutable SetPartitions{T}
+    s::AbstractVector{T}
+end
+
+length(s::SetPartitions) = npartitions(s)
+
+partitions(s::AbstractVector) = SetPartitions(s)
+
+start(p::SetPartitions) = (n = length(p.s); (zeros(Int32, n), ones(Int32, n-1), n, 1))
+done(p::SetPartitions, s) = !isempty(s) && s[1][1] > 0
+next(p::SetPartitions, s) = nextsetpartition(p.s, s...)
+
+function nextsetpartition(s, a, b, n, m)
+    function makeparts(s, a, m)
+        part = {}
+        for i = 0:m
+            ss = s[findin(a,i)]
+            if !isempty(ss)
+                push!(part, ss)
+            end
+        end
+        part
     end
-    # H5
-    if j == 1
-      break # terminate
+
+    if isempty(s);  return ({s}, ([1], Int[], n, 1));  end
+
+    part = makeparts(s,a,m)
+
+    if a[end] != m
+        a[end] += 1
+    else
+        local j
+        for j = n-1:-1:1
+            if a[j] != b[j]
+                break
+            end
+        end
+        a[j] += 1
+        m = b[j] + (a[j] == b[j])
+        for k = j+1:n-1
+            a[k] = 0
+            b[k] = m
+        end
+        a[end] = 0
     end
-    a[j] = a[j] + 1
-    # H6
-    m = b[j] + (a[j] == b[j])
-    j = j + 1
-    while j < n
-      a[j] = 0
-      b[j] = m
-      j = j + 1
+
+    return (part, (a,b,n,m))
+
+end
+
+
+npartitions(p::SetPartitions) = nsetpartitions(length(p.s))
+npartitions(v::AbstractVector) = nsetpartitions(length(v))
+
+const _nsetpartitions = (Int=>Int)[]
+function nsetpartitions(n::Int)
+    if n < 0
+        0
+    elseif n < 2
+        1
+    elseif (wn = get(_nsetpartitions, n, 0)) > 0
+        wn
+    else
+        wn = 0
+        for k = 0:n-1
+            wn += binomial(n-1,k)*nsetpartitions(n-1-k)
+        end
+        _nsetpartitions[n] = wn
     end
-    a[n] = 0
-  end
 end
 
+
 # For a list of integers i1, i2, i3, find the smallest 
 #     i1^n1 * i2^n2 * i3^n3 >= x
 # for integer n1, n2, n3

base/deprecated.jl

@@ -376,3 +376,7 @@ function addprocs_sge(np::Integer)
     error("Base.addprocs_sge is discontinued - add package ClusterManagers and then use ClusterManagers.addprocs_sge instead.")
 end
 export addprocs_sge
+
+function integer_partitions(n,m)
+    error("integer_partitions(n,m) has been renamed to partitions(n,m), and is now an iterator.  Please update your code.")
+end

base/exports.jl

@@ -517,6 +517,8 @@ export
     ndims,
     nnz,
     nonzeros,
+    npartitions,
+    nsetpartitions,
     nthperm!,
     nthperm,
     ones,

doc/stdlib/base.rst

@@ -3156,7 +3156,7 @@ Combinatorics
 
 .. function:: reverse(v [, start=1 [, stop=length(v) ]] )
 
-   Reverse vector ``v``, optionally from start to stop.
+   Return a copy of ``v`` reversed from start to stop.
 
 .. function:: reverse!(v [, start=1 [, stop=length(v) ]]) -> v
 
@@ -3176,20 +3176,45 @@ Combinatorics
    iterator object. Use ``collect(permutations(a,n))`` to get an array
    of all permutations.
 
-.. function:: integer_partitions(n, m)
+.. function:: partitions(n)
+
+   Generate all integer arrays that sum to ``n``. Because the number of
+   partitions can be very large, this function returns an iterator
+   object. Use ``collect(partitions(n))`` to get an array of all
+   partitions.
+
+.. function:: partitions(n, m)
 
    Generate all arrays of ``m`` integers that sum to ``n``. Because
-   the number of partitions can be very large, this function runs inside
-   a Task to produce values on demand. Write
-   ``c = @task integer_partitions(n,m)``, then iterate ``c`` or call
-   ``consume`` on it.
+   the number of partitions can be very large, this function returns an
+   iterator object. Use ``collect(partitions(n,m))`` to get an array of
+   all partitions.
 
 .. function:: partitions(array)
 
-   Generate all set partitions of the elements of an array, represented as
-   arrays of arrays. Because the number of partitions can be very large, this
-   function runs inside a Task to produce values on demand. Write
-   ``c = @task partitions(a)``, then iterate ``c`` or call ``consume`` on it.
+   Generate all set partitions of the elements of an array,
+   represented as arrays of arrays. Because the number of partitions
+   can be very large, this function returns an iterator object. Use
+   ``collect(partitions(array))`` to get an array of all partitions.
+
+.. function:: npartitions(n)
+
+   Calculate the number of ways of writing the integer ``n`` as a sum
+   of positive integers.
+
+.. function:: npartitions(n, m)
+
+   Calculate the number of ways of writing the integer ``n`` as a sum
+   of ``m`` positive integers.
+
+.. function:: npartitions(array)
+
+   Calculate the number of ways to partition ``array``, assuming
+   unique elements.  Equivalent to ``nsetpartitions(length(array))``
+
+.. function:: nsetpartitions(n)
+
+   Calculate the number of ways to partition a set of size ``n``.
 
 
 Statistics

test/Makefile

@@ -6,7 +6,7 @@ TESTS = all core keywordargs numbers strings unicode collections hashing	\
 	math functional bigint sorting statistics spawn parallel arpack file	\
 	git pkg pkg2 resolve resolve2 suitesparse complex version pollfd mpfr	\
 	broadcast socket floatapprox priorityqueue readdlm regex float16        \
-	combinations
+	combinatorics
 
 $(TESTS) ::
 	@$(PRINT_JULIA) $(call spawn,$(JULIA_EXECUTABLE)) ./runtests.jl $@