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    <p>The following will build a binary tree, and then analyze it and extract the results:</p> <pre><code>Clear[intParts]; intParts[num_, elems_List] /; Total[elems] &lt; num := p[]; intParts[num_, {fst_, rest___}] /; fst &lt; num := {p[fst, intParts[num - fst, {rest}]], intParts[num, {rest}]}; intParts[num_, {fst_, rest___}] /; fst &gt; num := intParts[num, {rest}]; intParts[num_, {num_, rest___}] := {pf[num], intParts[num, {rest}]}; Clear[nextPosition]; nextPosition = Compile[{{pos, _Integer, 1}}, Module[{ctr = 0, len = Length[pos]}, While[ctr &lt; len &amp;&amp; pos[[len - ctr]] == 1, ++ctr]; While[ctr &lt; len &amp;&amp; pos[[len - ctr]] == 2, ++ctr]; Append[Drop[pos, -ctr], 1]], CompilationTarget -&gt; "C"]; Clear[getPartitionsFromTree, getPartitions]; getPartitionsFromTree[tree_] := Map[Extract[tree, #[[;; -3]] &amp;@FixedPointList[nextPosition, #]] &amp;, Position[tree, _pf, Infinity]] /. pf[x_] :&gt; x; getPartitions[num_, elems_List] := getPartitionsFromTree@intParts[num, Reverse@Sort[elems]]; </code></pre> <p>For example, </p> <pre><code>In[14]:= getPartitions[200,Prime~Array~150]//Short//Timing Out[14]= {0.5,{{3,197},{7,193},{2,5,193},&lt;&lt;4655&gt;&gt;,{3,7,11,13,17,19,23,29,37,41}, {2,3,5,11,13,17,19,23,29,37,41}}} </code></pre> <p>This is not insanely fast, and perhaps the algorithm could be optimized further, but at least the number of partitions does not grow as fast as for <code>IntegerPartitions</code>.</p> <p>Edit:</p> <p>It is interesting that simple memoization speeds the solution up about twice on the example I used before:</p> <pre><code>Clear[intParts]; intParts[num_, elems_List] /; Total[elems] &lt; num := p[]; intParts[num_, seq : {fst_, rest___}] /; fst &lt; num := intParts[num, seq] = {p[fst, intParts[num - fst, {rest}]], intParts[num, {rest}]}; intParts[num_, seq : {fst_, rest___}] /; fst &gt; num := intParts[num, seq] = intParts[num, {rest}]; intParts[num_, seq : {num_, rest___}] := intParts[num, seq] = {pf[num], intParts[num, {rest}]}; </code></pre> <p>Now, </p> <pre><code>In[118]:= getPartitions[200, Prime~Array~150] // Length // Timing Out[118]= {0.219, 4660} </code></pre>
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    1. POA variation of IntegerPartition?
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    1. USLeonid Shifrin
    1. USLeonid Shifrin
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    1. COThis appears to work nicely. It does not include the parameter for set length, but I am going to play with it a bit. Thank you.
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    2. COI wasn't able to easily constrain the length. The problem is that solutions are accumulated from any sub-branch of the tree, not necessarily from the root. The best thing would be to somehow not produce lots of "false" branches at all, plus constrain the lengths of branches at the time we build the tree, but I could not figure out a good way to do this.
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    3. CO@Leonid Shifrin I will need some time to understand the method you provided. I would like to try to implement it myself, but do you believe this method can be extended to Rational arguments?
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