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<DIV>I have now calculated one more term of each sequence, finding another error.</DIV>
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<DIV>The m=3 case starts:</DIV>
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<DIV>1,1,1,2,4,7,13,25,49.</DIV>
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<DIV>And the infinite dimensional case starts:</DIV>
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<DIV>1,1,1,2,4,7,14,28,58.</DIV>
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<DIV>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.</DIV>
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<DIV>Franklin T. Adams-Watters<BR> <BR>-----Original Message-----<BR>From: <A href="mailto:franktaw@netscape.net">franktaw@netscape.net</A><BR>
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<DIV>I did find an error in my calculations. The last term of each of these should be one larger. For m=3:</DIV>
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<DIV>1,1,1,2,4,7,13,24</DIV>
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<DIV>and for the infinite case:</DIV>
<DIV> </DIV>
<DIV>1,1,1,2,4,7,14,27</DIV>
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<DIV>Franklin T. Adams-Watters<BR> <BR>-----Original Message-----<BR>From: <A href="javascript:parent.ComposeTo('franktaw@netscape.net');">franktaw@netscape.net</A><BR></DIV>
<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><FONT face="Courier New"></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>
<DIV><FONT face="Courier New"></FONT> </DIV>
<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>
<DIV><FONT face="Courier New"></FONT> </DIV>
<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>
<DIV><FONT face="Courier New"></FONT> </DIV>
<DIV><FONT face="Courier New">1,1,1,2,4,7,14,26</FONT></DIV>
<DIV><FONT face="Courier New"></FONT> </DIV>
<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>
<DIV><FONT face="Courier New"></FONT> </DIV>
<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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