tasks a-s

Task/Ordered-Partitions/0DESCRIPTION

@@ -0,0 +1,38 @@
+In this task we want to find the ordered partitions into fixed-size blocks. This task is related to [[Combinations]] in that it has to do with discrete mathematics and moreover a helper function to compute combinations is (probably) needed to solve this task.
+
+<math>partitions(\mathit{arg}_1,\mathit{arg}_2,...,\mathit{arg}_n)</math> should generate all distributions of the elements in <math>\{1,...,\Sigma_{i=1}^n\mathit{arg}_i\}</math> into <math>n</math> blocks of respective size <math>\mathit{arg}_1,\mathit{arg}_2,...,\mathit{arg}_n</math>.
+
+Example 1: <math>partitions(2,0,2)</math> would create:
+
+<pre>
+{({1, 2}, {}, {3, 4}),
+ ({1, 3}, {}, {2, 4}),
+ ({1, 4}, {}, {2, 3}),
+ ({2, 3}, {}, {1, 4}),
+ ({2, 4}, {}, {1, 3}),
+ ({3, 4}, {}, {1, 2})}
+</pre>
+
+Example 2: <math>partitions(1,1,1)</math> would create:
+
+<pre>
+{({1}, {2}, {3}),
+ ({1}, {3}, {2}),
+ ({2}, {1}, {3}),
+ ({2}, {3}, {1}),
+ ({3}, {1}, {2}),
+ ({3}, {2}, {1})}
+</pre>
+
+Note that the number of elements in the list is
+:<math>{\mathit{arg}_1+\mathit{arg}_2+...+\mathit{arg}_n \choose \mathit{arg}_1} \cdot {\mathit{arg}_2+\mathit{arg}_3+...+\mathit{arg}_n \choose \mathit{arg}_2} \cdot \ldots \cdot {\mathit{arg}_n \choose \mathit{arg}_n}</math>
+(see [http://en.wikipedia.org/wiki/Binomial_coefficient the definition of the binomial coefficient] if you are not familiar with this notation) and the number of elements remains the same regardless of how the argument is permuted
+(i.e. the [http://en.wikipedia.org/wiki/Multinomial_coefficient multinomial coefficient]). Also, <math>partitions(1,1,1)</math> creates the permutations of <math>\{1,2,3\}</math> and thus there would be <math>3! = 6</math> elements in the list.
+
+Note: Do not use functions that are not in the standard library of the programming language you use. Your file should be written so that it can be executed on the command line and by default outputs the result of <math>partitions(2,0,2)</math>. If the programming language does not support polyvariadic functions pass a list as an argument.
+
+'''Notation'''
+
+Remarks on the used notation for the task in order to understand it easierly.
+
+<math>\{1, \ldots, n\}</math> denotes the set of consecutive numbers from <math>1</math> to <math>n</math>, e.g. <math>\{1,2,3\}</math> if <math>n = 3</math>. <math>\Sigma</math> is the mathematical notation for summation, e.g. <math>\Sigma_{i=1}^3 i = 6</math> (see also [http://en.wikipedia.org/wiki/Summation#Capital-sigma_notation]). <math>\mathit{arg}_1,\mathit{arg}_2,...,\mathit{arg}_n</math> are the arguments — natural numbers — that the sought function receives.

Task/Ordered-Partitions/1META.yaml

@@ -0,0 +1,2 @@
+---
+note: Discrete math

Task/Ordered-Partitions/Ada/ordered-partitions-1.ada

@@ -0,0 +1,13 @@
+with Ada.Containers.Indefinite_Ordered_Sets;
+with Ada.Containers.Ordered_Sets;
+package Partitions is
+   -- Argument type for Create_Partitions: Array of Numbers
+   type Arguments is array (Positive range <>) of Natural;
+   package Number_Sets is new Ada.Containers.Ordered_Sets
+     (Natural);
+   type Partition is array (Positive range <>) of Number_Sets.Set;
+   function "<" (Left, Right : Partition) return Boolean;
+   package Partition_Sets is new Ada.Containers.Indefinite_Ordered_Sets
+     (Partition);
+   function Create_Partitions (Args : Arguments) return Partition_Sets.Set;
+end Partitions;

Task/Ordered-Partitions/Ada/ordered-partitions-2.ada

@@ -0,0 +1,148 @@
+package body Partitions is
+   -- compare number sets (not provided)
+   function "<" (Left, Right : Number_Sets.Set) return Boolean is
+      use type Ada.Containers.Count_Type;
+      use Number_Sets;
+      Left_Pos  : Cursor := Left.First;
+      Right_Pos : Cursor := Right.First;
+   begin
+      -- compare each element, until one or both lists finishes
+      while Left_Pos /= No_Element and then Right_Pos /= No_Element loop
+         -- compare elements
+         if Element (Left_Pos) < Element (Right_Pos) then
+            return True;
+         elsif Element (Left_Pos) > Element (Right_Pos) then
+            return False;
+         end if;
+         -- increase iterator
+         Next (Left_Pos);
+         Next (Right_Pos);
+      end loop;
+      -- Right is longer
+      if Right_Pos /= No_Element then
+         return True;
+      else
+         -- Left is longer, or Left and Right are identical.
+         return False;
+      end if;
+   end "<";
+   -- compare two Partitions
+   function "<" (Left, Right : Partition) return Boolean is
+      use type Ada.Containers.Count_Type;
+      use type Number_Sets.Set;
+   begin
+      -- check length
+      if Left'Length < Right'Length then
+         return True;
+      elsif Left'Length > Right'Length then
+         return False;
+      end if;
+      -- same length
+      if Left'Length > 0 then
+         for I in Left'Range loop
+            if Left (I) < Right (I) then
+               return True;
+            elsif Left (I) /= Right (I) then
+               return False;
+            end if;
+         end loop;
+      end if;
+      -- length = 0 are always smallest
+      return False;
+   end "<";
+   -- create partitions (as the task describes)
+   function Create_Partitions (Args : Arguments) return Partition_Sets.Set is
+      -- permutations needed
+      type Permutation is array (Positive range <>) of Natural;
+      -- exception to be thrown after last permutation reached
+      No_More_Permutations : exception;
+      -- get initial permutation (ordered small->big)
+      function Initial_Permutation (Max : Natural) return Permutation is
+         Result : Permutation (1 .. Max);
+      begin
+         for I in 1 .. Max loop
+            Result (I) := I;
+         end loop;
+         return Result;
+      end Initial_Permutation;
+      -- get next permutation
+      function Next_Permutation (Current : Permutation) return Permutation is
+         K      : Natural     := Current'Last - 1;
+         L      : Positive    := Current'Last;
+         Result : Permutation := Current;
+      begin
+         -- 1. Find the largest index k such that a[k] < a[k + 1].
+         while K /= 0 and then Current (K) > Current (K + 1) loop
+            K := K - 1;
+         end loop;
+         -- If no such index exists, the permutation is the last permutation.
+         if K = 0 then
+            raise No_More_Permutations;
+         end if;
+         -- 2. Find the largest index l such that a[k] < a[l].
+         -- Since k + 1 is such an index, l is well defined
+         -- and satisfies k < l.
+         while Current (K) > Current (L) loop
+            L := L - 1;
+         end loop;
+         -- 3. Swap a[k] with a[l].
+         Result (K) := Current (L);
+         Result (L) := Current (K);
+         -- 4. Reverse the sequence from a[k + 1] up to and including the
+         -- final element a[n].
+         for I in 1 .. (Result'Last - K) / 2 loop
+            declare
+               Temp : constant Natural := Result (K + I);
+            begin
+               Result (K + I)               := Result (Result'Last - I + 1);
+               Result (Result'Last - I + 1) := Temp;
+            end;
+         end loop;
+         return Result;
+      end Next_Permutation;
+      Result : Partition_Sets.Set;
+      Sum    : Natural := 0;
+   begin
+      -- get number of elements
+      for I in Args'Range loop
+         Sum := Sum + Args (I);
+      end loop;
+      declare
+      -- initial permutation
+         Current_Permutation : Permutation := Initial_Permutation (Sum);
+      begin
+         -- loop through permutations
+         loop
+         -- create Partition (same count of Number_Sets.Set as Args)
+            declare
+               Item              : Natural := Current_Permutation'First;
+               Current_Partition : Partition (Args'Range);
+            begin
+               -- loop each partition
+               for I in Args'Range loop
+                  -- fill in the number of elements requested
+                  for J in 1 .. Args (I) loop
+                     Current_Partition (I).Insert
+                       (New_Item => Current_Permutation (Item));
+                     Item := Item + 1;
+                  end loop;
+               end loop;
+               -- insert partition into result set
+               Result.Insert (New_Item => Current_Partition);
+            exception
+               when Constraint_Error =>
+                  -- partition was already inserted, ignore it.
+                  -- this happens when one of the args > 1.
+                  null;
+            end;
+            -- create next permutation
+            Current_Permutation := Next_Permutation (Current_Permutation);
+         end loop;
+      exception
+         when No_More_Permutations =>
+            -- no more permutations, we are finished
+            null;
+      end;
+      return Result;
+   end Create_Partitions;
+end Partitions;

Task/Ordered-Partitions/Ada/ordered-partitions-3.ada

@@ -0,0 +1,61 @@
+with Ada.Text_IO;
+with Partitions;
+procedure Main is
+   package Natural_IO is new Ada.Text_IO.Integer_IO (Natural);
+   Example_Partitions : Partitions.Partition_Sets.Set;
+begin
+   Ada.Text_IO.Put_Line ("Partitions for (2, 0, 2):");
+   -- create partition
+   Example_Partitions := Partitions.Create_Partitions (Args => (2, 0, 2));
+   -- pretty print the result
+   declare
+      use type Partitions.Partition_Sets.Cursor;
+      Position : Partitions.Partition_Sets.Cursor := Example_Partitions.First;
+   begin
+      Ada.Text_IO.Put ('{');
+      while Position /= Partitions.Partition_Sets.No_Element loop
+         if Position /= Example_Partitions.First then
+            Ada.Text_IO.Put (' ');
+         end if;
+         declare
+            Current_Partition : constant Partitions.Partition :=
+               Partitions.Partition_Sets.Element (Position);
+         begin
+            Ada.Text_IO.Put ('(');
+            for I in Current_Partition'Range loop
+               Ada.Text_IO.Put ('{');
+               declare
+                  use type Partitions.Number_Sets.Cursor;
+                  Current_Number : Partitions.Number_Sets.Cursor :=
+                     Current_Partition (I).First;
+               begin
+                  while Current_Number /= Partitions.Number_Sets.No_Element
+                  loop
+                     Natural_IO.Put
+                       (Item  =>
+                           Partitions.Number_Sets.Element (Current_Number),
+                        Width => 1);
+                     Partitions.Number_Sets.Next (Current_Number);
+                     if Current_Number /=
+                        Partitions.Number_Sets.No_Element then
+                        Ada.Text_IO.Put (',');
+                     end if;
+                  end loop;
+               end;
+               Ada.Text_IO.Put ('}');
+               if I /= Current_Partition'Last then
+                  Ada.Text_IO.Put (", ");
+               end if;
+            end loop;
+         end;
+         Ada.Text_IO.Put (')');
+         Partitions.Partition_Sets.Next (Position);
+         if Position /= Partitions.Partition_Sets.No_Element then
+            Ada.Text_IO.Put (',');
+            Ada.Text_IO.New_Line;
+         end if;
+      end loop;
+      Ada.Text_IO.Put ('}');
+      Ada.Text_IO.New_Line;
+   end;
+end Main;

Task/Ordered-Partitions/BBC-BASIC/ordered-partitions.bbc

@@ -0,0 +1,52 @@
+      DIM list1%(2) : list1%() = 2, 0, 2
+      PRINT "partitions(2,0,2):"
+      PRINT FNpartitions(list1%())
+      DIM list2%(2) : list2%() = 1, 1, 1
+      PRINT "partitions(1,1,1):"
+      PRINT FNpartitions(list2%())
+      DIM list3%(3) : list3%() = 1, 2, 0, 1
+      PRINT "partitions(1,2,0,1):"
+      PRINT FNpartitions(list3%())
+      END
+
+      DEF FNpartitions(list%())
+      LOCAL i%, j%, n%, p%, o$, x%()
+      n% = DIM(list%(),1)
+      DIM x%(SUM(list%())-1)
+      FOR i% = 0 TO n%
+        IF list%(i%) THEN
+          FOR j% = 1 TO list%(i%)
+            x%(p%) = i%
+            p% += 1
+          NEXT
+        ENDIF
+      NEXT i%
+      REPEAT
+        FOR i% = 0 TO n%
+          o$ += " ( "
+          FOR j% = 0 TO DIM(x%(),1)
+            IF x%(j%) = i% o$ += STR$(j%+1) + " "
+          NEXT
+          o$ += ")"
+        NEXT i%
+        o$ += CHR$13 + CHR$10
+      UNTIL NOT FNperm(x%())
+      = o$
+
+      DEF FNperm(x%())
+      LOCAL i%, j%
+      FOR i% = DIM(x%(),1)-1 TO 0 STEP -1
+        IF x%(i%) < x%(i%+1) EXIT FOR
+      NEXT
+      IF i% < 0 THEN = FALSE
+      j% = DIM(x%(),1)
+      WHILE x%(j%) <= x%(i%) j% -= 1 : ENDWHILE
+      SWAP x%(i%), x%(j%)
+      i% += 1
+      j% = DIM(x%(),1)
+      WHILE i% < j%
+        SWAP x%(i%), x%(j%)
+        i% += 1
+        j% -= 1
+      ENDWHILE
+      = TRUE

Task/Ordered-Partitions/C/ordered-partitions-1.c

@@ -0,0 +1,51 @@
+#include <stdio.h>
+
+int next_perm(int size, int * nums)
+{
+        int *l, *k, tmp;
+
+        for (k = nums + size - 2; k >= nums && k[0] >= k[1]; k--) {};
+        if (k < nums) return 0;
+
+        for (l = nums + size - 1; *l <= *k; l--) {};
+        tmp = *k; *k = *l; *l = tmp;
+
+        for (l = nums + size - 1, k++; k < l; k++, l--) {
+                tmp = *k; *k = *l; *l = tmp;
+        }
+
+        return 1;
+}
+
+void make_part(int n, int * sizes)
+{
+        int x[1024], i, j, *ptr, len = 0;
+
+        for (ptr = x, i = 0; i < n; i++)
+                for (j = 0, len += sizes[i]; j < sizes[i]; j++, *(ptr++) = i);
+
+        do {
+                for (i = 0; i < n; i++) {
+                        printf(" { ");
+                        for (j = 0; j < len; j++)
+                                if (x[j] == i) printf("%d ", j);
+
+                        printf("}");
+                }
+                printf("\n");
+        } while (next_perm(len, x));
+}
+
+int main()
+{
+        int s1[] = {2, 0, 2};
+        int s2[] = {1, 2, 3, 4};
+
+        printf("Part 2 0 2:\n");
+        make_part(3, s1);
+
+        printf("\nPart 1 2 3 4:\n");
+        make_part(4, s2);
+
+        return 1;
+}

Task/Ordered-Partitions/C/ordered-partitions-2.c

@@ -0,0 +1,51 @@
+#include <stdio.h>
+
+typedef unsigned int uint;
+
+int parts[] = {2, 1, 2};
+#define n_parts sizeof(parts)/sizeof(parts[0])
+int bits[n_parts];
+
+void show_part(uint x)
+{
+	uint i;
+	putchar('{');
+	for (i = 0; (1 << i) <= x; i ++)
+		if (x & (1 << i)) printf(" %d", i + 1);
+
+	printf("%s", " } ");
+}
+
+void gen_bits(uint mask, uint all, uint res, int n, int pid)
+{
+	uint i;
+	while (!n) {
+		bits[pid++] = res;
+		if (pid == n_parts) {
+			for (i = 0; i < n_parts; i++)
+				show_part(bits[i]);
+			putchar('\n');
+			return;
+		}
+		mask = all &= ~res;
+		res = 0;
+		n = parts[pid];
+	}
+
+	while (mask) {
+		mask &= ~(i = mask & -(int)mask);
+		gen_bits(mask, all, res | i, n - 1, pid);
+	}
+}
+
+int main(void)
+{
+	uint i, m;
+	for (m = 1, i = 0; i < n_parts; i++)
+		m <<= parts[i];
+	m--;
+
+	gen_bits(m, m, 0, parts[0], 0);
+
+	return 0;
+}

Task/Ordered-Partitions/Common-Lisp/ordered-partitions.lisp

@@ -0,0 +1,46 @@
+(defun fill-part (x i j l)
+  (let ((e (elt x i)))
+    (loop for c in l do
+	    (loop while (>= j (length e)) do
+		  (setf j 0 e (elt x (incf i))))
+	    (setf (elt e j) c)
+	    (incf j))))
+
+;;; take a list of lists and return next partitioning
+;;; it's caller's responsibility to ensure each sublist is sorted
+(defun next-part (list cmp)
+  (let* ((l (coerce list 'vector))
+	 (i (1- (length l)))
+	 (e (elt l i)))
+    (loop while (<= 0 (decf i)) do
+	  ;; e holds all the right most elements
+	  (let ((p (elt l i)) (q (car (last e))))
+	    ;; find the right-most list that has an element that's smaller
+	    ;; than _something_ in later lists
+	    (when (and p (funcall cmp (first p) q))
+	      ;; find largest element that can be increased
+	      (loop for j from (1- (length p)) downto 0 do
+		    (when (funcall cmp (elt p j) q)
+		      ;; find the smallest element that's larger than
+		      ;; that largest
+		      (loop for x from 0 to (1- (length e)) do
+			    (when (funcall cmp (elt p j) (elt e x))
+			      (rotatef (elt p j) (elt e x))
+			      (loop while (< (incf j) (length p)) do
+			      	(setf (elt p j) (elt e (incf x))
+				      (elt e x) nil))
+			      (fill-part l i j (remove nil e))
+			      (return-from next-part l))))
+		    (setf e (append e (list (elt p j))))))
+	    (setf e (append e p))))))
+
+(let ((a '#((1 2) () (3 4))))
+  (loop while a do
+	(format t "~a~%" a)
+	(setf a (next-part a #'<))))
+
+(write-line "with dupe elements:")
+(let ((a '#((a c) (c c d))))
+  (loop while a do
+	(format t "~a~%" a)
+	(setf a (next-part a #'string<))))

Task/Ordered-Partitions/D/ordered-partitions.d

@@ -0,0 +1,34 @@
+import std.stdio, std.algorithm, std.range, std.array, std.conv;
+// http://rosettacode.org/wiki/Combinations#D
+import combinations4: Comb;
+
+alias int[] iRNG;
+
+iRNG setDiff(iRNG s, iRNG c) {
+    return setDifference(s, c).array();
+}
+
+iRNG[][] orderPart(iRNG blockSize...) {
+    iRNG sum = iota(1, 1 + blockSize.reduce!q{a + b}()).array();
+
+    iRNG[][] p(iRNG s, in iRNG b) {
+        if (b.length == 0)
+            return [[]];
+        iRNG[][] res;
+        foreach (c; Comb.On(s, b[0]))
+            foreach (r; p(setDiff(s, c), b[1 .. $]))
+                res ~= c.dup ~ r;
+        return res;
+    }
+
+    return p(sum, blockSize);
+}
+
+void main(string[] args) {
+    auto b = args.length > 1 ?
+        args[1 .. $].map!(to!int)().array() :
+        [2, 0, 2];
+
+    foreach (p; orderPart(b))
+        writeln(p);
+}

Task/Ordered-Partitions/GAP/ordered-partitions.gap

@@ -0,0 +1,28 @@
+FixedPartitions := function(arg)
+	local aux;
+	aux := function(i, u)
+		local r, v, w;
+		if i = Size(arg) then
+			return [[u]];
+		else
+			r := [ ];
+			for v in Combinations(u, arg[i]) do
+				for w in aux(i + 1, Difference(u, v)) do
+					Add(r, Concatenation([v], w));
+				od;
+			od;
+			return r;
+		fi;
+	end;
+	return aux(1, [1 .. Sum(arg)]);
+end;
+
+
+FixedPartitions(2, 0, 2);
+# [ [ [ 1, 2 ], [  ], [ 3, 4 ] ], [ [ 1, 3 ], [  ], [ 2, 4 ] ],
+#   [ [ 1, 4 ], [  ], [ 2, 3 ] ], [ [ 2, 3 ], [  ], [ 1, 4 ] ],
+#   [ [ 2, 4 ], [  ], [ 1, 3 ] ], [ [ 3, 4 ], [  ], [ 1, 2 ] ] ]
+
+FixedPartitions(1, 1, 1);
+# [ [ [ 1 ], [ 2 ], [ 3 ] ], [ [ 1 ], [ 3 ], [ 2 ] ], [ [ 2 ], [ 1 ], [ 3 ] ],
+#   [ [ 2 ], [ 3 ], [ 1 ] ], [ [ 3 ], [ 1 ], [ 2 ] ], [ [ 3 ], [ 2 ], [ 1 ] ] ]

Task/Ordered-Partitions/Go/ordered-partitions.go

@@ -0,0 +1,60 @@
+package main
+
+import (
+	"fmt"
+	"os"
+	"strconv"
+)
+
+func gen_part(n, res []int, pos int) {
+	if pos == len(res) {
+		x := make([][]int, len(n))
+		for i, c := range res {
+			x[c] = append(x[c], i+1)
+		}
+
+		fmt.Println(x)
+		return
+	}
+
+	for i := range n {
+		if n[i] == 0 {
+			continue
+		}
+		n[i], res[pos] = n[i]-1, i
+		gen_part(n, res, pos+1)
+		n[i]++
+	}
+}
+
+func ordered_part(n_parts []int) {
+	fmt.Println("Ordered", n_parts)
+
+	sum := 0
+	for _, c := range n_parts {
+		sum += c
+	}
+
+	gen_part(n_parts, make([]int, sum), 0)
+}
+
+func main() {
+	if len(os.Args) < 2 {
+		ordered_part([]int{2, 0, 2})
+		return
+	}
+	n := make([]int, len(os.Args)-1)
+	var err error
+	for i, a := range os.Args[1:] {
+		n[i], err = strconv.Atoi(a)
+		if err != nil {
+			fmt.Println(err)
+			return
+		}
+		if n[i] < 0 {
+			fmt.Println("negative partition size not meaningful")
+			return
+		}
+	}
+	ordered_part(n)
+}

Task/Ordered-Partitions/Groovy/ordered-partitions-1.groovy

@@ -0,0 +1,13 @@
+def partitions = { int... sizes ->
+    int n = (sizes as List).sum()
+    def perms = n == 0 ? [[]] : (1..n).permutations()
+    Set parts = perms.collect { p -> sizes.collect { s -> (0..<s).collect { p.pop() } as Set } }
+    parts.sort{ a, b ->
+        if (!a) return 0
+        def comp = [a,b].transpose().find { it[0] != it[1] }
+        if (!comp) return 0
+        def recomp = comp.collect{ it as List }.transpose().find { it[0] != it[1] }
+        if (!recomp) return 0
+        return recomp[0] <=> recomp[1]
+    }
+}

Task/Ordered-Partitions/Groovy/ordered-partitions-2.groovy

@@ -0,0 +1,3 @@
+partitions(2, 0, 2).each {
+    println it
+}

Task/Ordered-Partitions/Haskell/ordered-partitions-1.hs

@@ -0,0 +1,13 @@
+import Data.List ((\\))
+
+comb :: Int -> [a] -> [[a]]
+comb 0 _      = [[]]
+comb _ []     = []
+comb k (x:xs) = map (x:) (comb (k-1) xs) ++ comb k xs
+
+partitions :: [Int] -> [[[Int]]]
+partitions xs = p [1..sum xs] xs
+    where p _ []      = [[]]
+          p xs (k:ks) = [ cs:rs | cs <- comb k xs, rs <- p (xs \\ cs) ks ]
+
+main = print $ partitions [2,0,2]

Task/Ordered-Partitions/Haskell/ordered-partitions-2.hs

@@ -0,0 +1,12 @@
+comb :: Int -> [a] -> [([a],[a])]
+comb 0 xs     = [([],xs)]
+comb _ []     = []
+comb k (x:xs) = [ (x:cs,zs) | (cs,zs) <- comb (k-1) xs ] ++
+                [ (cs,x:zs) | (cs,zs) <- comb  k    xs ]
+
+partitions :: [Int] -> [[[Int]]]
+partitions xs = p [1..sum xs] xs
+    where p _ []      = [[]]
+          p xs (k:ks) = [ cs:rs | (cs,zs) <- comb k xs, rs <- p zs ks ]
+
+main = print $ partitions [2,0,2]

Task/Ordered-Partitions/Haskell/ordered-partitions-3.hs

@@ -0,0 +1,15 @@
+-- choose m out of n items, return tuple of chosen and the rest
+choose aa _ 0 = [([], aa)]
+choose aa@(a:as) n m
+	| n == m = [(aa, [])]
+	| otherwise =	map (\(x,y) -> (a:x, y)) (choose as (n-1) (m-1)) ++
+			map (\(x,y) -> (x, a:y)) (choose as (n-1) m)
+
+partitions x = combos [1..n] n x where
+	n = sum x
+	combos _ _ [] = [[]]
+	combos s n (x:xs) = [ l : r |	(l,rest) <- choose s n x,
+					r <- combos rest (n - x) xs]
+
+
+main = mapM_ print $ partitions [5,5,5]

Task/Ordered-Partitions/J/ordered-partitions-1.j

@@ -0,0 +1,2 @@
+require'stats'
+partitions=: ([,] {L:0 (i.@#@, -. [)&;)/"1@>@,@{@({@comb&.> +/\.)

Task/Ordered-Partitions/J/ordered-partitions-2.j

@@ -0,0 +1,39 @@
+   partitions 2 0 2
+┌───┬┬───┐
+│0 1││2 3│
+├───┼┼───┤
+│0 2││1 3│
+├───┼┼───┤
+│0 3││1 2│
+├───┼┼───┤
+│1 2││0 3│
+├───┼┼───┤
+│1 3││0 2│
+├───┼┼───┤
+│2 3││0 1│
+└───┴┴───┘
+   partitions 1 1 1
+┌─┬─┬─┐
+│0│1│2│
+├─┼─┼─┤
+│0│2│1│
+├─┼─┼─┤
+│1│0│2│
+├─┼─┼─┤
+│1│2│0│
+├─┼─┼─┤
+│2│0│1│
+├─┼─┼─┤
+│2│1│0│
+└─┴─┴─┘
+   #partitions 2 3 5
+2520
+   #partitions 5 7 11
+|out of memory: partitions
+|   #    partitions 5 7 11
+   */ (! +/\.)5 7 11
+1070845776
+   #partitions 3 5 7
+360360
+   */ (! +/\.)3 5 7
+360360

Task/Ordered-Partitions/Lua/ordered-partitions.lua

@@ -0,0 +1,89 @@
+--- Create a list {1,...,n}.
+local function range(n)
+  local res = {}
+  for i=1,n do
+    res[i] = i
+  end
+  return res
+end
+
+--- Return true if the element x is in t.
+local function isin(t, x)
+  for _,x_t in ipairs(t) do
+    if x_t == x then return true end
+  end
+  return false
+end
+
+--- Return the sublist from index u to o (inclusive) from t.
+local function slice(t, u, o)
+  local res = {}
+  for i=u,o do
+    res[#res+1] = t[i]
+  end
+  return res
+end
+
+--- Compute the sum of the elements in t.
+-- Assume that t is a list of numbers.
+local function sum(t)
+  local s = 0
+  for _,x in ipairs(t) do
+    s = s + x
+  end
+  return s
+end
+
+--- Generate all combinations of t of length k (optional, default is #t).
+local function combinations(m, r)
+  local function combgen(m, n)
+    if n == 0 then coroutine.yield({}) end
+    for i=1,#m do
+      if n == 1 then coroutine.yield({m[i]})
+      else
+        for m0 in coroutine.wrap(function() combgen(slice(m, i+1, #m), n-1) end) do
+          coroutine.yield({m[i], unpack(m0)})
+        end
+      end
+    end
+  end
+  return coroutine.wrap(function() combgen(m, r) end)
+end
+
+--- Generate a list of partitions into fized-size blocks.
+local function partitions(...)
+  local function helper(s, ...)
+    local args = {...}
+    if #args == 0 then return {% templatetag openvariable %}{% templatetag closevariable %} end
+    local res = {}
+    for c in combinations(s, args[1]) do
+      local s0 = {}
+      for _,x in ipairs(s) do if not isin(c, x) then s0[#s0+1] = x end end
+      for _,r in ipairs(helper(s0, unpack(slice(args, 2, #args)))) do
+        res[#res+1] = {{unpack(c)}, unpack(r)}
+      end
+    end
+    return res
+  end
+  return helper(range(sum({...})), ...)
+end
+
+-- Print the solution
+io.write "["
+local parts = partitions(2,0,2)
+for i,tuple in ipairs(parts) do
+  io.write "("
+  for j,set in ipairs(tuple) do
+    io.write "{"
+    for k,element in ipairs(set) do
+      io.write(element)
+      if k ~= #set then io.write(", ") end
+    end
+    io.write "}"
+    if j ~= #tuple then io.write(", ") end
+  end
+  io.write ")"
+  if i ~= #parts then io.write(", ") end
+end
+io.write "]"
+io.write "\n"

Task/Ordered-Partitions/Mathematica/ordered-partitions-1.math

@@ -0,0 +1,4 @@
+w[partitions_]:=Module[{s={},t=Total@partitions,list=partitions,k}, n=Length[list];
+    While[n>0,s=Join[s,{Take[t,(k=First[list])]}];t=Drop[t,k];list=Rest[list];n--]; s]
+
+m[p_]:=(Sort/@#)&/@(w[#,p]&/@Permutations[Range@Total[p]])//Union

Task/Ordered-Partitions/Mathematica/ordered-partitions-2.math

@@ -0,0 +1,3 @@
+Grid@m[{2, 0, 2}]
+
+Grid@m[{1, 1, 1}]

Task/Ordered-Partitions/Perl-6/ordered-partitions.pl6

@@ -0,0 +1,10 @@
+sub partition(@mask is copy) {
+    my $last = [+] @mask or return [[] xx @mask];
+    sort gather for @mask.kv -> $k,$v {
+        next unless $v;
+        temp @mask[$k] -= 1;
+        for partition @mask { .take.[$k].push($last) }
+    }
+}
+
+.perl.say for partition [2,0,2];

Task/Ordered-Partitions/Perl/ordered-partitions-1.pl

@@ -0,0 +1,43 @@
+use Thread 'async';
+use Thread::Queue;
+
+sub make_slices {
+        my ($n, @avail) = (shift, @{ +shift });
+
+        my ($q, @part, $gen);
+        $gen = sub {
+                my $pos = shift;        # where to start in the list
+                if (@part == $n) {
+                        # we accumulated enough for a partition, emit them and
+                        # wait for main thread to pick them up, then back up
+                        $q->enqueue(\@part, \@avail);
+                        return;
+                }
+
+                # obviously not enough elements left to make a partition, back up
+                return if (@part + @avail < $n);
+
+                for my $i ($pos .. @avail - 1) {                # try each in turn
+                        push @part, splice @avail, $i, 1;       # take one
+                        $gen->($i);                             # go deeper
+                        splice @avail, $i, 0, pop @part;        # put it back
+                }
+        };
+
+        $q = new Thread::Queue;
+        (async{ &$gen;                  # start the main work load
+                $q->enqueue(undef)      # signal that there's no more data
+        })->detach;     # let the thread clean up after itself, not my problem
+
+        return $q;
+}
+
+my $qa = make_slices(4, [ 0 .. 9 ]);
+while (my $a = $qa->dequeue) {
+        my $qb = make_slices(2, $qa->dequeue);
+
+        while (my $b = $qb->dequeue) {
+                my $rb = $qb->dequeue;
+                print "@$a | @$b | @$rb\n";
+        }
+}

Task/Ordered-Partitions/Perl/ordered-partitions-2.pl

@@ -0,0 +1,48 @@
+sub partitions {
+        my $sum = 0;
+        $sum += $_ for @_;              # total number of elements
+        make_part (     $_[-1],         # desired partition size
+                        0,              # initial trial position
+                        [ (0) x $sum ], # table recording of used element
+                        [],             # current pick for current partition
+                        [       $#_,    # total number of partitions
+                                \@_,    # partition sizes
+                                []      # for output, each partition's elements
+                        ]               # Note: last group of args wrapped in array ref
+        );                              #       to reduce argument passing overhead
+}
+
+sub make_part {
+        my ($n, $pos, $used, $picked, $more) = @_;
+        return if $pos > @$used;
+
+        # the making-next-partition part
+        if (!$n) {
+                my ($part_idx, $sizes, $q) = @$more;
+                push @$q, $picked;
+                if ($part_idx > 1) {
+                        make_part($sizes->[$part_idx-1], 0, $used, [],
+                                        [ $part_idx-1, $sizes, $q]);
+                } else {
+                        my @x = grep { !$used->[$_] } 0 .. (@$used-1);
+                        print "[ @$_ ]" for @$q;
+                        print "[ @x ]\n";
+                }
+                pop @$q;
+                return;
+        }
+
+        # the picking-element-to-make-partition part
+        for my $i ($pos .. @$used - 1) {
+                next if $used->[$i];
+                push @$picked, $i;
+                $used->[$i] = 1;
+
+                make_part($n - 1, $i + 1, $used, $picked, $more);
+
+                $used->[$i] = 0;
+                pop @$picked;
+        }
+}
+
+partitions(4, 2, 4);

Task/Ordered-Partitions/PicoLisp/ordered-partitions.l

@@ -0,0 +1,11 @@
+(de partitions (Args)
+   (let Lst (range 1 (apply + Args))
+      (recur (Args Lst)
+         (ifn Args
+            '(NIL)
+            (mapcan
+               '((L)
+                  (mapcar
+                     '((R) (cons L R))
+                     (recurse (cdr Args) (diff Lst L)) ) )
+               (comb (car Args) Lst) ) ) ) ) )

Task/Ordered-Partitions/Python/ordered-partitions-1.py

@@ -0,0 +1,15 @@
+from itertools import combinations
+
+def partitions(*args):
+    def p(s, *args):
+        if not args: return [[]]
+        res = []
+        for c in combinations(s, args[0]):
+            s0 = [x for x in s if x not in c]
+            for r in p(s0, *args[1:]):
+                res.append([c] + r)
+        return res
+    s = range(sum(args))
+    return p(s, *args)
+
+print partitions(2, 0, 2)

Task/Ordered-Partitions/Python/ordered-partitions-2.py

@@ -0,0 +1,10 @@
+from itertools import combinations as comb
+
+def partitions(*args):
+    def minus(s1, s2): return [x for x in s1 if x not in s2]
+    def p(s, *args):
+        if not args: return [[]]
+        return [[c] + r for c in comb(s, args[0]) for r in p(minus(s, c), *args[1:])]
+    return p(range(1, sum(args) + 1), *args)
+
+print partitions(2, 0, 2)

Task/Ordered-Partitions/Tcl/ordered-partitions-1.tcl

@@ -0,0 +1,52 @@
+package require Tcl 8.5
+package require struct::set
+
+# Selects all k-sized combinations from a list.
+# "Borrowed" from elsewhere on RC
+proc selectCombinationsFrom {k l} {
+    if {$k == 0} {return {}} elseif {$k == [llength $l]} {return [list $l]}
+    set all {}
+    set n [expr {[llength $l] - [incr k -1]}]
+    for {set i 0} {$i < $n} {} {
+        set first [lindex $l $i]
+	incr i
+        if {$k == 0} {
+            lappend all $first
+	} else {
+	    foreach s [selectCombinationsFrom $k [lrange $l $i end]] {
+		lappend all [list $first {*}$s]
+	    }
+        }
+    }
+    return $all
+}
+
+# Construct the partitioning of a given list
+proc buildPartitions {lst n args} {
+    # Base case when we have no further partitions to process
+    if {[llength $args] == 0} {
+	return [list [list $lst]]
+    }
+    set result {}
+    set c [selectCombinationsFrom $n $lst]
+    if {[llength $c] == 0} {set c [list $c]}
+    foreach comb $c {
+	# Sort necessary for "nice" order
+	set rest [lsort -integer [struct::set difference $lst $comb]]
+	foreach p [buildPartitions $rest {*}$args] {
+	    lappend result [list $comb {*}$p]
+	}
+    }
+    return $result
+}
+
+# Wrapper that assembles the initial list and calls the partitioner
+proc partitions args {
+    set sum [tcl::mathop::+ {*}$args]
+    set startingSet {}
+    for {set i 1} {$i <= $sum} {incr i} {
+	lappend startingSet $i
+    }
+
+    return [buildPartitions $startingSet {*}$args]
+}

Task/Ordered-Partitions/Tcl/ordered-partitions-2.tcl

@@ -0,0 +1,4 @@
+puts [partitions 1 1 1]
+puts [partitions 2 2]
+puts [partitions 2 0 2]
+puts [partitions 2 2 0]

Task/Ordered-Partitions/Ursala/ordered-partitions-1.ursala

@@ -0,0 +1,8 @@
+#import std
+#import nat
+
+opart =
+
+-+
+  ~&art^?\~&alNCNC ^|JalSPfarSPMplrDSL/~& ^DrlPrrPlXXS/~&rt ^DrlrjXS/~&l choices@lrhPX,
+  ^\~& nrange/1+ sum:-0+-

Task/Ordered-Partitions/Ursala/ordered-partitions-2.ursala

@@ -0,0 +1,3 @@
+#cast %nLLL
+
+test = opart <2,0,2>