[seqfan] Re: Generalization of A005646

[Andrew Weimholt]

On Fri, Jan 1, 2010 at 3:24 PM, Robert Munafo <mrob27 at gmail.com> wrote:
> You say "more than 2-way partitions are allowed".
>
> Is it also required that each partition is 3-way, or can it be a mix of
> partitions of 2-way and/or 3-way?
>
> (If the former, I suppose the value would be 0 for N=1, *undefined* for N=2,
> and 1 for N-3)
>

I am pretty sure Franklin intended the latter definition, as I get the
same initial terms
when computing by hand.

> Franklin T. Adams-Watters wrote:
>
>> I'm interested in generalizing A005646, and the associated triangle, to the
>> case where more than 2-way partitions are allowed in making the
>> classification.  We would want to require that each partition be completely
>> essential; that is, combining any two parts in any partition would result in
>> at least one pair of points being indistinguishable.
>>
>> I don't have enough terms to even try looking this up to see if it is in
>> the OEIS; I believe that it starts (from n = 1):
>>
>> 1,1,2,5
>>
>> If it is present, it has neither a link to nor from A005646.
>>
>> More terms, anyone?  Is there any literature on this question?

The taxonomies of the generalized sequence can all be generated from
the taxonomies of
A005646 by combining sets of two or more binary partitions, and then
re-collapsing sub-partitions as needed.
For example, if combining two binary partitions, a 3-way or 4-way
partition will be created.
In some cases, the 4-way partition must be collapsed back into a 3-way
partition to satisfy the "completely essential" requirement.

Andrew