[seqfan] Partition encoding and conjugation: A122111, weird correspondence, or is it just obvious?

[Antti Karttunen]

Cheers,

First we need a tabf-table like:
http://oeis.org/A112798
but organized by this principle:
http://oeis.org/wiki/User:Antti_Karttunen/Mail_by_Marc_LeBrun_Re_Partition_ordering_on_SeqFan_list_posted_12_Jan_2006

it should begin as
row
 2    1;
 3    1, 1;
 4    2;
 5    1, 1, 1;
 6    2, 2;
 7    1, 1, 1, 1;
 8    3;
 9    1, 2;
10   2, 2, 2;
11   1, 1, 1, 1, 1;
12   3, 3;
13   1, 1, 1, 1, 1, 1;
14   2, 2, 2, 2;
15   1, 2, 2;
16   4;

It's not yet in OEIS:
http://oeis.org/search?q=1%2C+1%2C1%2C+2%2C+1%2C1%2C1%2C+2%2C2%2C+1%2C1%2C1%2C1%2C+3%2C+1%2C2%2C+2%2C2%2C2%2C+1%2C1%2C1%2C1%2C1%2C+&sort=&language=&go=Search

Now, the funny thing, is that it seems that
http://oeis.org/A122111
can be defined _both ways_, as conjugation in A112798 encoding system
and as conjugation in Marc's "funny system", as I have claimed in this
comment
a(n) = A075158(A122111(1+A075157(n)) - 1).
in http://oeis.org/A129594
and also in a few comments in
http://oeis.org/A129595

But is it really true? Either I hadn't really read Franklin's description
in http://oeis.org/A122111 when I superficially checked my claims
(and apparently they have matched with the sixty terms given in
A122111, because I usually do these kind of empirical sanity checks
before submitting such claims. But just now I don't have any platform
which would handle prime factorization easily)...,
or then, maybe I saw the obviousness of this all in 2007, but have
grown steadily dumber then?

When I started rethinking this last summer, I first thought that only
primes and powers of two map nicely to each other, and thus by jumping
between those two systems and doing conjugation at the partition stage
between, we would get a permutation of natural numbers which would fix
primes and powers of 2, but otherwise be quite wild, but if what is
said above is really true, then this "wild permutation" reduces to
A000027, which is not so wild after all...

In any case, the "Bulgarian operation" (see http://oeis.org/A226062 )
should be different in those two systems.


Yours,

Antti Karttunen