[seqfan] Re: Extension of A005646

franktaw at netscape.net franktaw at netscape.net
Sat Dec 19 13:33:26 CET 2009

Oh, sorry; I see now that these are for unlabeled objects.  I was 
confused by Andrew's list of labeled partitions.

For n odd, you can avoid problems by always having the first part 
bigger than the second; but for even n this doesn't always work.  I 
assume that you are doing, at some point, permutation of the objects to 
see if two classifications are the same (presumably with 
optimizations); I think you have to try flipping any balanced binary 
partitions in that phase as well.  (There are of course optimizations 
to be made here; for example looking at the objects that occur alone, 
put more of these in the first part if possible.)

I would also like to see a triangle with T(n,k) being the number of 
classifications of n objects using k binary partitions.

This starts:

0 1
0 0 1
0 0 1 2
0 0 0 3 3 (I think)
0 0 0 3 17 6 (from Andrew's list)

Franklin T. Adams-Watters

-----Original Message-----
From: franktaw at netscape.net

To avoid this, simply always include a particular number (0 if you
start with it, else 1) in the first part of each partition.

Franklin T. Adams-Watters

-----Original Message-----
From: Robert Munafo <mrob27 at gmail.com>
I suspect that I am treating two classifications as distinct when
complementing one of more of their constituent binary partitions would
make them equivalent, but not clear on how to detect or avoid.

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