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    <p>While probably not the most elegant solution, this would work:</p> <pre><code>def f(n, s): def g(n,s): if n == 1: yield (s,) else: for i in xrange(s + 1): for j in g(n - 1,s - i): yield (i,) + j L = list(g(3, 5)) D = [] i = 1 while i != len(L): for j in xrange(len(L[i])): if abs(L[i][j] - L[i-1][j]) &gt; 1: D.append(L.pop(i)) i -= 1 break for d in D: ins = True for j in xrange(len(L[-1])): if abs(L[-1][j] - d[j]) &gt; 1: ins = False break if ins: L.append(d) D.remove(d) i += 1 return L for i in f(3, 5): print i </code></pre> <p>This prints:</p> <pre><code>(0, 0, 5) (0, 1, 4) (0, 2, 3) (0, 3, 2) (0, 4, 1) (0, 5, 0) (1, 4, 0) (2, 3, 0) (3, 2, 0) (4, 1, 0) (5, 0, 0) (4, 0, 1) (3, 0, 2) (2, 0, 3) (1, 0, 4) (1, 1, 3) (2, 1, 2) (3, 1, 1) (2, 2, 1) (1, 2, 2) (1, 3, 1) </code></pre> <p>It basically defines <code>g</code> inside of <code>f</code>, as the generator to create the permuations. It then goes through the list of that and checks for each element if the difference (the <code>abs</code> part) between the element in the one sub-list and the one before (not explained very well, I know... But you get it) is greater than 1, and if so removes that list, appends it to <code>D</code> and reduces the index by 1 (which is why I used <code>while</code> instead of <code>for</code>).</p> <p><strong>Edit:</strong> After every check of an element, it goes through <code>D</code> and sees if anything fits to <code>L</code>, and if so, appends it.</p> <p>It then returns the filtered list.</p>
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    1. POHow to order permutations so that at least 1 element in each permutation differs by exactly 1
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    1. COThis solution is compliant with the requirement that a least one element in each row has a delta of 1 from any adjacent row; however, we lost valuable permutations when we popped, but did not reinsert. A proper algorithm will generate output exactly as shown in the "Desired output" block in the problem description.
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    2. CO@BrianMullen: Didn't see the desired output, updated post to fix that :)
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    3. COWe don't lose any permutations. But, order of the permutations in the output matters. In the "desired output" the left most element is always increasing and elements to the right always smoothly vacillate from one extreme to another (without causing the sum to exceed S). If these permutations are used as inputs to a function, the outputs of the function should also generate a some smooth wave form (which is my goal). If the discrete output from my function can be modeled as a smooth wave form, it will be easier for my algorithm to find local maximums (gradient ascent).
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