fyi - recently submitted to OEIS (but not posted yet) sequences

[Richard Mathar]

There is also some ambiguity in the question you are asking: do you 
want
to count the number of ways to subpartition, or the number of distinct
sums for the parts?  I will assume for the moment that it is the former.

Note next that [1,1,2,2] is not the partition with the smallest sum 
that
achieves this bound; [1,1,1,1] (which I prefer to write [1^4]) also 
does:
   [1^4]
   [1,1] [1,1]
   [1] [1] [1] [1]
and in general, we can always use [1^m] to achieve tau(m) parts.  In
particular, this shows that a(4) <= 6.

This is not usually the best way to do it, however; the partition
[1^4,2^4] can be partitioned into equal parts 7 ways:
   [1^4,2^4]
   [1^4,2] [2^3]
   [1,1,2,2] [1,1,2,2]
   [1^4] [2,2] [2,2]
   [1,1,2] [1,1,2] [2,2]
   [1,2] [1,2] [1,2] [1,2]
   [1,1] [1,1] [2] [2] [2] [2]
This shows that a(7) <= 8.  The smallest number with tau(n) >= 7 is 24.

We can also show that a(5) <= 7, using the partition [1^3,2^3,3].

So we have upper bounds:
1,2,4,6,7,8,8
where the first 3 values are known to be correct.  If these values are 
all
correct, the sequence is not in the OEIS.

Franklin T. Adams-Watters

-----Original Message-----
From: David W. Wilson <wilson.d at anseri.com>

The partition (1, 1, 2, 2) can be split into either 1, 2, or
3 subpartitions having equal sum, to wit:

     (1, 1, 2, 2)
     (1, 2), (1, 2)
     (1, 1), (2), (2)

This is not possible for any smaller nonempty partition. If
we define a(n) as the smallest number of elements in a partition that 
can be
split into 1 through n equal-sum subpartitions, this example shows a(3) 
= 4.

Can we find a few small a(n)? Is the sequence already in the
OEIS?






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