[seqfan] Re: A019318 ? (Sorry - figured it out.)

[r.shepherd]

{I see just now that A019318 has already been revised;  much of
the following still applies though}.

From: "Brendan McKay" <bdm at cs.anu.edu.au>
Sent: Tuesday, May 27, 2003 3:52 AM

>
> I agree with you about "queens", that really must go.  I suggest
> "Inequivalent ways of choosing n squares of an n x n chessboard,
> considering rotations and reflections."
>

...but why do away with queens without doing away with the
chessboard also?  More generally, another suggestion would be
"Distinct ways of placing n identical objects in an n x n array,
disregarding rotations and reflections."  (or choosing, if you
prefer).  "considering" confuses me;  don't you mean "disregarding"?
(I'm not sure what standard terminology would be, though, and
I'm still counting positions to make sure.).  Also, of course,
same thing can be said about n^2-n objects.

FWIW, I think the application of the results to a chessboard
is natural  (and should, therefore, still be mentioned):  If one is
trying to solve a non-attacking pieces problem, this result helps
make sure no positions are overlooked (or counted twice).

Even more generally, is a corresponding result for n x n x n
in the OEIS?  Does a similarly nice formula exist  (My
memory fails me about Burnside's lemma) for 3-D (and
beyond)?


> It should be a piece of cake to find an exact formula using
> Burnside's Lemma, but I'll give someone else the pleasure.
>

As Jim Nastos mentioned, the author of the sequence,
Mario Velucchi, gave a formula.  Using it, later today I
plan to submit a simple PARI program and some more terms
of the sequence that I found last night.

It'll look something like this:

p(a,b,N)=if(N%2==0,(a+b)^(N^2)+2*(a+b)^N*\
(a^2+b^2)^((N^2-N)/2)+3*(a^2+b^2)^(N^2/2)+\
2*(a^4+b^4)^(N^2/4),\
(a+b)^(N^2)+2*(a+b)*(a^4+b^4)^((N^2-1)/4)+\
(a+b)*(a^2+b^2)^((N^2-1)/2)+\
4*(a+b)^N*(a^2+b^2)^((N^2-N)/2))

for(k=1,20,print1(polcoeff(p(a,1,k),k)/8,","))

1,2,16,252,6814,244344,10746377,553319048,32611596056,2163792255680,
159593799888052,12952412056879996,1147044793316531040,
110066314584030859544,11375695977099383509351,
1259843950257390597789296,148842380543159458506703546,
18685311541775061906510072648,2483858381692984848273972297368,
348545122958862200122401771463328,

General question:  Does the absence of the "more" keyword
on a sequence mean that more terms are not desired?

> Brendan.

Regards,
Rick Shepherd