Conjugate m-dimensional partitions

[franktaw at netscape.net]

I have now calculated one more term of each sequence, finding another error.
 
The m=3 case starts:
 
1,1,1,2,4,7,13,25,49.
 
And the infinite dimensional case starts:
 
1,1,1,2,4,7,14,28,58.
 
While these are still hand-calculated, I have now checked them against the total number of m-dimensional partitions, looking at the number of symmetries for each, so I am now confident that these values are correct.  These are definitely not in the OEIS; I will submit them.
 
Franklin T. Adams-Watters
 
-----Original Message-----
From: franktaw at netscape.net

I did find an error in my calculations.  The last term of each of these should be one larger.  For m=3:
 
1,1,1,2,4,7,13,24
 
and for the infinite case:
 
1,1,1,2,4,7,14,27
 
Franklin T. Adams-Watters
 
-----Original Message-----
From: franktaw at netscape.net

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.)
 
...
 
For m = 3, I believe the sequence starts (this is hand-calculated):
 
1,1,1,2,4,7,13,23
 
There are a couple sequences with 2,4,7,13,23; I don't think either of them is this one, but I can't be sure.
 
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.  I believe that the number of infinite-dimensional partitions up to conjugacy starts:
 
1,1,1,2,4,7,14,26
 
Again, I can't be sure that this isn't in the OEIS, but I don't think so.
 
Can somebody verify and extend these?  Any other comments?
 
Franklin T. Adams-Watters



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