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Re: Partitioning a list into unequal partitions

  • To: mathgroup at
  • Subject: [mg47546] Re: Partitioning a list into unequal partitions
  • From: Paul Abbott <paul at>
  • Date: Thu, 15 Apr 2004 03:41:04 -0400 (EDT)
  • Organization: The University of Western Australia
  • References: <c5gfdp$an0$> <c5j8aj$r4t$>
  • Sender: owner-wri-mathgroup at

In article <c5j8aj$r4t$1 at>,
 Paul Abbott <paul at> wrote:

> In article <c5gfdp$an0$1 at>, "DIAMOND Mark R." <dot at> 
> wrote:
> > Could someone please show me a simple (non-procedural) way of partitioning a
> > list into 1,2,3 ... n disjoint sublists, where the length of the list is
> > guaranteed to be correct (i.e. n*(n+1)/2)
> If I understand your question (a definite example would have helped), 
> for a list, lst, how about
>   Sort[ReplaceList[lst, {___, a__, ___} -> {a}]]
> This gives a list of length n*(n+1)/2 if lst is of length n.

It is clear now that I did not understand your question!
> > I can't see a way, and yet I'm sure there *must* be a one-liner using Fold.
> I don't see how Fold will do what you want. 

Here are three ways (slightly) different to those posted thus far:

Define two utility functions:

  tri[n_] := (Sqrt[8 n + 1] - 1)/2
  TriangularQ[n_] := IntegerQ[tri[n]]

[1] Use FoldList for indexing:

 TriangularList[l_List] := With[{n = tri[Length[l]]}, Take[l, #]& /@ 
     FoldList[Plus, {1, 1}, Transpose[{Range[n - 1], Range[2, n]}]]]

[2] Use a nested Range along with FoldList:

  TriangularList[l_List] :=  Module[{n = tri[Length[l]]},     
    l[[#]] & /@ (Range[Range[n]] + FoldList[Plus, 0, Range[n - 1]])]

[1] and [2] will truncate a list to the nearest triangular-sized list.

[3] Use Split with a counter:

 TriangularList[l_List] := Module[{i=0},Split[l,Not[TriangularQ[++i]]&]]

This will put any extra items into the last list in the output.

Are you trying to pack a list into an upper or lower triangular matrix? 
If so you could use SparseArray

  lst = CharacterRange["a", "f"];

  SparseArray[Flatten[Table[{i,j},{i,tri[Length[lst]]},{j,i}],1] -> lst]

  {{"a", 0, 0}, {"b", "c", 0}, {"d", "e", "f"}}

  DeleteCases[%, 0, 2]
  {{"a"}, {"b", "c"}, {"d", "e", "f"}}


Paul Abbott                                   Phone: +61 8 9380 2734
School of Physics, M013                         Fax: +61 8 9380 1014
The University of Western Australia      (CRICOS Provider No 00126G)         
35 Stirling Highway
Crawley WA 6009                      mailto:paul at 

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