[seqfan] BITXOR Partitions

[Marc LeBrun]

 >=Paul D Hanna
 > Could someone extend the following sequence of BITXOR partitions?
 > I do not think that sequence is already in the OEIS.
 > I have not submitted due to lack of terms with certainty.
 > BITXOR partitions: number of partitions of n into distinct integers, 
each being less than or equal to n,
 > under the binary exclusive OR operation, without regard to order.

This is already in the OEIS as A054243.

 > Sequence begins (I think):
 > 1,1,2,1,2,5,6,1,...

Actually, it begins (from the empty partition a(0)=1)
1,1,1,2,2,4,8,16,16,32,64,128,256,512,1024,2048,2048,4096,...

 > bitxor partitions of 2^n equals 1.
 > ...

Too low!  This misses, for example, all the ones you get by appending 
"partitions of zero", as in 4 = 1 + 2 + 3 + 4, and so forth.

 > One could also define the partitions of n under the binary AND operation 
(if not already in OEIS).

I think that's in there too, as well as IOR, 'though I don't have time to 
run them all down right now...


Indulge me to mention that this is an instance of what I like to call 
"numbral theory": whenever you have a set of indexed objects that you can 
do some kind of arithmetic on, then the indexes act as "shadows" of the 
objects, and you can generally talk about lots of analogs, such as 
partitions, primes, even generating functions, etc.  It would be great to 
systematically "fill out" the entries for as many of these systems as 
possible in the OEIS (which sometimes finds unexpected "hits" with existing 
sequences)!