Conjugate m-dimensional partitions

[A.N.W.Hone]

Dear seqfans,

Please ignore my guess about rational solutions of the functional
equation

A(x)=A(-x A(x)).     ---- (1)

I have proved that if a meromorphic solution has a pole at x=c for some
non-zero c, then there can be only one such c.

So far the only explicit solutions I have are

A(x) = -c / (x-c),

A^*(x) = 1 - c/x,

and P(x) = -c(x-b)/x/(x-c);

only the first one is regular at x=0. The functional equation (1) also
admits the involution A(x) -> 1/A(1/x).

Since Franklin has now pointed out that this functional equation is
not relevant to the sequence being considered, I will not bother
seqfans with any further comments on (1) (unless it should reappear for
some other reason!).

Andy

On Thu, 11 May 2006 franktaw at netscape.net wrote:

> I have now calculated one more term of each sequence, finding another error.
>
> The m=3 case starts:
>
> 1,1,1,2,4,7,13,25,49.
>
> And the infinite dimensional case starts:
>
> 1,1,1,2,4,7,14,28,58.
>
> While these are still hand-calculated, I have now checked them against the total number of m-dimensional partitions, looking at the number of symmetries for each, so I am now confident that these values are correct.  These are definitely not in the OEIS; I will submit them.
>
> Franklin T. Adams-Watters
>
> -----Original Message-----
> From: franktaw at netscape.net
>
> I did find an error in my calculations.  The last term of each of these should be one larger.  For m=3:
>
> 1,1,1,2,4,7,13,24
>
> and for the infinite case:
>
> 1,1,1,2,4,7,14,27
>
> Franklin T. Adams-Watters
>
> -----Original Message-----
> From: franktaw at netscape.net
>
> An m-dimensional partition can be considered as a set of points in an (m+1) dimensional corner.  (This is a generalized Ferrers diagram.)  As such, each is one of (m+1)! conjugate partitions induced by permutation of the axes.  (Some of these may be the same, of course.)
>
> ...
>
> For m = 3, I believe the sequence starts (this is hand-calculated):
>
> 1,1,1,2,4,7,13,23
>
> There are a couple sequences with 2,4,7,13,23; I don't think either of them is this one, but I can't be sure.
>
> We can also ask how many infinite-dimensional partitions there are up to conjugacy.  There are infinitely many infinite-dimensional partitions of any n >= 2, but any m-dimensional partition of n can be reduced to at most n-2 dimensions by conjugacy.  I believe that the number of infinite-dimensional partitions up to conjugacy starts:
>
> 1,1,1,2,4,7,14,26
>
> Again, I can't be sure that this isn't in the OEIS, but I don't think so.
>
> Can somebody verify and extend these?  Any other comments?
>
> Franklin T. Adams-Watters
>
>
>
> Try the New Netscape Mail Today!
> Virtually Spam-Free | More Storage | Import Your Contact List
> http://mail.netscape.com
>
>
> Try the New Netscape Mail Today!
> Virtually Spam-Free | More Storage | Import Your Contact List
> http://mail.netscape.com
> ___________________________________________________
> Try the New Netscape Mail Today!
> Virtually Spam-Free | More Storage | Import Your Contact List
> http://mail.netscape.com
>