Conjugate m-dimensional partitions

[franktaw at netscape.net]

I have submitted that table as pre-numbered sequence A119269.  It looks 
like you have a few more values; please extend it once it's out there 
(which should be soon).

Your three "guessed" values are correct, assuming the accuracy of your 
other numbers.  The final increments in the columns become fixed (as 
you conjecture) once you get to twice the dimension; in reverse order, 
1, 2, 6, 19, and (from your numbers) 60.  This could be A052544, but I 
have no particular reason to think it is.

Franklin T. Adams-Watters

-----Original Message-----
From: Max <maxale at gmail.com>

  On 5/8/06, franktaw at netscape.net <franktaw at netscape.net> wrote:

  > An m-dimensional partition can be considered as a set of points in 
an (m+1)
  > dimensional corner. (This is a generalized Ferrers diagram.) As 
such, each
  > is one of (m+1)! conjugate partitions induced by permutation of the 
axes.
 > (Some of these may be the same, of course.)
 >
  > So, how many m-dimensional partitions are there, up to conjugacy? 
For m =
 > 1, we have A046682:
 >
 > 1,1,1,2,3,4,6,8,12,16,22,29,40,...
 >
 > and for m = 2, A000786:
 >
 > [1,]1,1,2,4,6,11,19,33,55,...
 >
 > For m = 3, I believe the sequence starts (this is hand-calculated):
 >
 > 1,1,1,2,4,7,13,23

 I've got the following table:

 m=0: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
 m=1: 1, 1, 1, 2, 3, 4, 6, 8, 12, 16, 22, 29, 40, 52, 69, 90, 118,
 151, 195, 248, 317
 m=2: 1, 1, 1, 2, 4, 6, 11, 19, 33, 55, 95, 158, 267, 442, 731,
 1193, 1947, 3137, 5039, 8026, 12726
 m=3: 1, 1, 1, 2, 4, 7, 13, 25, 49, 93, 181, 351, 687, 1332, 2591,
 5003, 9644, 18462, 35208, 66721, 125840
 m=4: 1, 1, 1, 2, 4, 7, 14, 27, 55, 111, 232, 486, 1039, 2226, 4820,
 10449, 22727
  m=5: 1, 1, 1, 2, 4, 7, 14, 28, 57, 117, 251, 543, 1209, 2724, 6251, 
14505
 m=6: 1, 1, 1, 2, 4, 7, 14, 28, 58, 119, 257, 562, 1268, 2910, 6844
 m=7: 1, 1, 1, 2, 4, 7, 14, 28, 58, 120, 259, 568, 1287, 2970
 m=8: 1, 1, 1, 2, 4, 7, 14, 28, 58, 120, 260

  > We can also ask how many infinite-dimensional partitions there are 
up to
  > conjugacy. There are infinitely many infinite-dimensional partitions 
of any
  > n >= 2, but any m-dimensional partition of n can be reduced to at 
most n-2
 > dimensions by conjugacy.

 In other words, this sequence is given by the diagonal T(m,m+2) in the
 table above and it happens to be:
 1, 1, 1, 2, 4, 7, 14, 28, 58, 120, 260, 571, 1296, 2998 (the last
 three values are guessed)

 It is also interesting to notice that
 T(m,m+3)=T(m,m+2)-1
 T(m,m+4)=T(m+1,m+4)-2=T(m+2,m+4)-3
 T(m,m+5)=T(m+1,m+5)-6 for m>=2
 T(m,m+6)=T(m+1,m+6)-19 for m>=4
 Perhaps, there exists a simple proof for that.

 It may also happen that for any fixed k, T(m+1,m+k)-T(m,m+k)=C(k) for
 all large enough m where C(k) is a constant depending on k but not on
 m. If so, for k=1..6 we have
 C(k): 0,0,1,2,6,19

 Max


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