[seqfan] Re: Unexpected help by A052515

[Rainer Rosenthal]

Rick Shepherd schrieb:
> Here is a very straightforward (using Ralf Stephan's claim) proof.
> [nice proof skipped, thanks!]
> 
> Ralf Stephan stated that, for n>2, A052515(n) = 2^n-2n-2.  Assuming
> that is true, then also a = 2*A052515(n).

I was able to understand Ralf's claim and prove it by simply
enumerating the pairs [t,complement(t)]. This yields 2^n, as there
are 2^n subsets t of {1,...,n}. We have to exclude the empty set
and its complement as well as the n singletons {1}, ..., {n}
and their complements. Thus we arrive at 2^n - 2n - 2 quite
easily.

It took a while until I realized that the sets in the pairs are
supposed to be complementary to each other. And I suggest to
put this piece of information into the definition:

  A052515  Number of pairs of complementary sets
                              =============
           of cardinality at least 2.

Best regards,
Rainer