Trac #20008: Implement non-recursive iterator for compositions

Compositions currently uses a recursive iterator, which also has extra
overhead of converting between the element classes.

URL: http://trac.sagemath.org/20008
Reported by: tscrim
Ticket author(s): Travis Scrimshaw
Reviewer(s): Darij Grinberg

src/sage/combinat/composition.py

@@ -1960,13 +1960,40 @@ def __iter__(self):
             sage: Compositions(0).list()
             [[]]
         """
-        if self.n == 0:
-            yield self.element_class(self, [])
-            return
+        for c in composition_iterator_fast(self.n):
+            yield self.element_class(self, c)
+
+def composition_iterator_fast(n):
+    """
+    Iterator over compositions of ``n`` yielded as lists.
+
+    TESTS::
 
-        for i in range(1,self.n+1):
-            for c in Compositions_n(self.n-i):
-                yield self.element_class(self, [i]+list(c))
+        sage: from sage.combinat.composition import composition_iterator_fast
+        sage: L = list(composition_iterator_fast(4)); L
+        [[1, 1, 1, 1], [1, 1, 2], [1, 2, 1], [1, 3], [2, 1, 1], [2, 2], [3, 1], [4]]
+        sage: type(L[0])
+        <type 'list'>
+    """
+    # Special cases
+    if n < 0:
+        return
+    if n == 0:
+        yield []
+        return
+
+    s = Integer(0) # Current sum
+    cur = [Integer(0)]
+    while cur:
+        cur[-1] += 1
+        s += 1
+        # Note that because we are adding 1 every time,
+        #   we will never have s > n
+        if s == n:
+            yield list(cur)
+            s -= cur.pop()
+        else:
+            cur.append(Integer(0))
 
 from sage.structure.sage_object import register_unpickle_override
 register_unpickle_override('sage.combinat.composition', 'Composition_class', Composition)