Add a multi-partition implementation to sage

[pabryan]

As discussed here[1], multi-partitions would be a useful addition to sage. I have an initial implementation soon to be uploaded.

[1]  http://groups.google.com/group/sage-combinat-devel/browse_thread/thread/aefcc71ad217d023/439360dc6b2641c5

[pabryan]

Implementation of multi-partitions

[ebriand]

Some comments:

• The functions for multipartitions do not seem to work correctly. Here is an incorrect (incomplete) result:

 
M=MultiPartitions([2,2])
for x in M.list(): print x
[[2, 2]]
[[1, 2], [1, 0]]
[[2, 1], [0, 1]]
[[1, 0], [1, 0], [0, 2]]
[[1, 1], [1, 0], [0, 1]]
[[2, 0], [0, 1], [0, 1]]
[[1, 0], [1, 0], [0, 1], [0, 1]]

Two multipartitions are missed: [[1,1],[1,1]] and [[2,0],[0,2]]

• I wonder whether MultiPartitions is a good name, because it seems it has a different meaning in a near context. See for instance this paper by G. Andrews:  http://es.scribd.com/doc/6625886/Multi-Partitions, or this other paper;  http://www.maths.qmul.ac.uk/~mf/papers/weight2.pdf, or these slides:  http://www.cs.clemson.edu/~goddard/MINI/2009/Swisher.pdf ."Vector partitions" may be safer (it is used for instance by George Andrews in his book on partitions).

• Should one represent the Multipartitions as lists of vectors with integer coordinates rather than lists of lists ? (because sometimes one has to do the sum of the parts).

• A few typos:
  • Line 6 of the file: "the" should not be repeated.
  • Line 9 of the file: replace "it's" with "its".
  • Line 63 of the file: replace "lexicographically order integer lists" with "lexicographically weakly decreasing lists of integers" or something better.
  • Also, one might replace the definition of multipartition as "equivalence class of tuples" (lines 4 to 10) with "multiset of tuples".