Partitions & Perseverance

[Wouter Meeussen]

Latest Records for C3 and C3v plane partitions:

C3 : 97 terms
{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 2, 2, 0, 3, 3, 0, 5,
6, 0, 7, 9, 0, 11, 16, 1, 14, 23, 2, 20, 36, 4, 27, 52, 7, 37, 78, 13, 48,
111, 21, 65, 163, 36, 83, 227, 56, 109, 322, 89, 139, 444, 135, 179, 618,
207, 226, 841, 305, 288, 1151, 453, 361, 1550, 656, 456, 2094, 950, 569,
2793, 1351, 712, 3730, 1922, 885, 4931, 2694, 1105, 6522, 3772, 1371, 8555,
5224, 1708, 11216, 7218, 2125, 14608, 9885, 2650} 

C3v : 151 terms
{1, 0, 0, 1, 0, 0, 2, 1, 0, 2, 1, 0, 2, 1, 0, 3, 2, 0, 4, 4, 0, 4, 5, 0, 5, 
7, 1, 6, 9, 1, 6, 11, 1, 8, 15, 2, 10, 20, 3, 10, 25, 4, 12, 33, 7, 14, 40, 
9, 15, 48, 12, 18, 60, 17, 20, 74, 23, 22, 89, 30, 26, 108, 40, 30, 130, 51, 
33, 157, 66, 37, 187, 85, 42, 222, 108, 47, 262, 136, 54, 311, 173, 62, 365, 
215, 68, 427, 268, 81, 500, 331, 93, 585, 406, 104, 680, 498, 122, 792, 612, 
142, 915, 742, 163, 1059, 900, 192, 1222, 1088, 225, 1407, 1311, 264, 1615, 
1573, 311, 1857, 1889, 371, 2123, 2252, 437, 2425, 2682, 523, 2770, 3187, 
623, 3162, 3779, 739, 3592, 4467, 885, 4086, 5277, 1063, 4633, 6206, 1266, 
5254, 7294, 1518, 5946, 8554, 1817, 6723, 10015, 2175, 7586, 11695, 2600} 

As you can imagine, some nifty programming was needed.
I tinkered with the Partitions[n,maxelement] function
to get a function that remembers previously calculated values.
Minimising storage space required a myPartitions[n,maxexactly]
function (producing the partitions of n with max-element exactly "maxexactly".
A similar function was built for the balanced partitions (first
element = number of elements).
Biggest intermediate storage requirement was 500 Mb.

Why do these sequences rise so slowly?
The requirement that the balanced (C3) or symmetric (C3v)
partitions need be stacked on an outer layer that majors (C3v)
or overmajors (C3) them keeps the number of combinations down.
In fact, a "majoring table" was set up, giving the indices of 
child-partitions for each parent-partition. Of course, there are
lots of grand-grand-parents...

Ferrers Scheme: overmajors= parent minus first row & column majors child
   parent:            child:
*  *  *  *  *  *
*  o  *  *  *  *    o  +  +  +
*  *  *  *          +  +
*  *  *             +  +
*  *                +
*

the C3 plane partition is made by rotating each partition 'round a C3 axis
to get:
outer onion peel:   inner peel:      stacked=C3
6  6  4  3  2  1                     6  6  4  3  2  1
5  1  1  1  1  1    4  2  2  1       5  5  3  3  2  1
4  1  1  1          3  1             4  4  2  1
3  1  1             1  1             3  2  2
2  1                1                2  2
2                                    2

but this operation is not required for counting.

For the C3v partitions, use was made of the symmetry of the partitions
by working with the upper diagonal part only:

6  6  3  2  2  2  becomes   6  5  1

+  +  +  +  +  +            +  +  +  +  +  +
*  +  +  +  +  +            +  +  +  +  +
*  *  +                     +
*  *
*  *  
*  *
(that's why just majoring, instead of over-majoring, is sufficient here)

Truely addicitve stuff. I'm cured now that it's written.
Maybe one day, someone will hit on a true GF for these sequences.

wouter.

Dr. Wouter L. J. MEEUSSEN
w.meeussen.vdmcc at vandemoortele.be
eu000949 at pophost.eunet.be