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
_______________________________________________ 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