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

[Antti Karttunen]

On Fri, Oct 18, 2013 at 12:04 PM, Antti Karttunen
<antti.karttunen at gmail.com> wrote:
> 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;

For A112798 we already have a sequence of "products of parts" as:
http://oeis.org/A003963
but for the above table not.

The partition sums seem not to be present:
http://oeis.org/search?q=1%2C2%2C2%2C3%2C4%2C4%2C3%2C3%2C6%2C5%2C6%2C6%2C8%2C5%2C4&sort=&language=&go=Search

(and if the conjecture that mapping between Marc's system and
A112798-system is equivalent to taking conjugations is true, then
those sums are for both of those tables, as conjugating doesn't change
the total sum of partition.
I wonder if there are any deeper significance/applications for that
correspondence?)

Similar related sequences, already computed for unordered partitions
encoded in the runlengths of binary expansion of n (Please see
crossrefs and comments in https://oeis.org/A227739 ) could (should?)
also be computed
for these two (or just one?) systems. Some of them should already be
present, because of their number-theoretical nature, e.g.
http://oeis.org/A055396
for the smallest part in the partitions encoded in A112798-system.


Cheers,

Antti


>
> 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