[seqfan] request for help with a sequence

[John Erickson]

First off, sorry I wasn't  sure how to reply directly to a thread so I just
used the same initial subject.

Second, sincere thanks for the replies, and indeed, I agree that the
unlabeled version is the most natural.

I however was interested in specific orbits under the canonical action of
S_n on {1,2,..,n}. In particular, I was interested in orbits that I called
"balanced" which are just orbits whose averages are all the same.
Fortunately,  A326512 about balanced partitions was already listed. A count
of all subgroups producing balanced orbits proved difficult, but the
special case of subgroups with the simplest balanced orbits of
{{1,2,...,n}} --the transitive subgroups--is tractable. Since the only tool
I was aware of was the Pyber inequality about A005432, I wanted to compare
labeled to labeled.

John