Conjugate m-dimensional partitions

[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.)
 
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
 
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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