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<DIV>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.)</DIV>
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<DIV>So, how many m-dimensional partitions are there, up to conjugacy? For m = 1, we have A046682:</DIV>
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<DIV><FONT face="Courier New">1,1,1,2,3,4,6,8,12,16,22,29,40,...</FONT></DIV>
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<DIV><FONT face="Courier New">and for m = 2, A000786:</FONT></DIV>
<DIV><FONT face="Courier New"></FONT> </DIV>
<DIV><FONT face="Courier New">[1,]1,1,2,4,6,11,19,33,55,...</FONT></DIV>
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<DIV><FONT face="Courier New">For m = 3, I believe the sequence starts (this is hand-calculated):</FONT></DIV>
<DIV><FONT face="Courier New"></FONT> </DIV>
<DIV><FONT face="Courier New">1,1,1,2,4,7,13,23</FONT></DIV>
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<DIV><FONT face="Courier New">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.</FONT></DIV>
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<DIV><FONT face="Courier New">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:</FONT></DIV>
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<DIV><FONT face="Courier New">1,1,1,2,4,7,14,26</FONT></DIV>
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<DIV><FONT face="Courier New">Again, I can't be sure that this isn't in the OEIS, but I don't think so.</FONT></DIV>
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<DIV><FONT face="Courier New">Can somebody verify and extend these? Any other comments?</FONT></DIV>
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<DIV>Franklin T. Adams-Watters<BR></DIV></DIV></DIV>
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