Fred, I remember well - and still have somewhere - your wonderful papers in the Computer Journal from 40 years ago, and those funny-smelling gray-colored foolscap notes you used to send me when I was collecting material for the 1973 Handbook (on Dedekind's problem, the postage stamp problem, etc.). The programs were brilliant then, you must have learned a lot of tricks in the mean time, plus you now have Magma. So I think I'm justified in being excited! Best regards Neil On Thu, Jan 9, 2014 at 5:56 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
[Hey --- not so much of the "old", young fella!]
I am keeping in mind to upload a counting program to OEIS, once the algorithm stabilises (whenever that happens, ho-ho!), along with (quite a few) more numbers.
But in any case, for the time being my involvement in this problem needs to avoid becoming too public (turning up collar and pulling hat down over face ...)
WFL
On 1/9/14, Neil Sloane <njasloane@gmail.com> wrote:
A Magma program from the old master? Of course we want it! Could you possibly add it to (say) A078099 ?
If it is a bit long, the usual thing is to put it in a file called a078099.txt (or a078099_MAGMA.txt) and upload it to go with the sequence in question.
Neil
On Thu, Jan 9, 2014 at 12:38 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
On reflection, a modicum of signposting might be in order: so note that colourings (of either class) number (2 or 3 times) T(m, n) = O(1+d)^(m n) .
Note also that the Maple counting program given under A078099 requires substantial editing, and is impracticably slow for m > 8 . Results from a more elaborate Magma program can be made available, should anyone feel sufficiently motivated to demand them ...
Fred Lunnon
On 1/9/14, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Within the (exiguous) bowels of OEIS A078099 there lurks a throwaway comment to the effect that 3-colouring the nodes of a m x n grid such that edge-adjacent pairs of nodes are assigned distinct colours is equivalent to 2-colouring the nodes such that no square (4-circuit) has both diagonals assigned distinct colours. [Establishing this equivalence is not entirely trivial.] In default of any earlier attribution, I propose to dub the latter colourings "Hardin" colourings.
Though I haven't sat down to prove it, it seems reasonable to assume that as n,m -> oo the density of diagonal pairs with equal colours approaches a limit d , for a `random' Hardin colouring. [We can just sum frequencies over distinct n x n colourings where n is large, then take means.]
I can show d > 0.5382 , and extrapolation suggests d ~ 0.5395 +/- 0.00005 ; can anybody improve on these values, find an explicit expression, or show a non-trivial upper bound for d ?
Fred Lunnon
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-- Dear Friends, I have now retired from AT&T. New coordinates:
Neil J. A. Sloane, President, OEIS Foundation 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Dear Friends, I have now retired from AT&T. New coordinates: Neil J. A. Sloane, President, OEIS Foundation 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com