Elithrion wrote:Also, an alternative "solution"! The orthogonality condition is actually pretty much equivalent to taking half the number of steps but being able to move freely (to visualise this, pretend you're moving diagonally), except for the very first walk, where with it you can get back to the origin and with half the steps you cannot. In every other case, though, the two options seem equivalent. So, if we replace orthogonality with shorter walks, we just get the need to count SAPs with no extra properties. And since they have a name, I assume there are already several papers out there that provide counts and maybe the formula is even known! *takes a bow*
Nathaniel wrote:(and in particular their value for 2n = 10 is less than our value for 8n = 40).
All of this seems to be an exercise in random walks, which would also seem to include the enumeration of many other extensions. Some of them consist of inserting multiple copies of a given segment, and account for such sequences as x, long x, long long x, and so on. It is not hard to visualize what they should look like, nor how many of them there are, although it is not necessarily profitable to continue listing them. But there is often a choice between possible insertions, even though the resulting figure is still one dimensional. For example, the polyhats, while one dimensional, admit a two-dimensional variation in which the height of the vertical bar in each segment may remain the same, increase by two, or decrease by two (although not to zero). The bracketing bars have to be stabilized, but this can generally be done with a variety of hooks, blocks, and snakes. A special case arises when some regularity is required, such as returning to an earlier height; this becomes the parenthesis balancing problem. Finally there is the truly two-dimensional problem of enumerating lakes, chains of ponds, or the mentioned quasars supposing that one has decided how the extension should take place. By and large, these are all questions which can be resolved by graph theory, and particularly through the algebra of the connection matrix. What might be interesting is to look at this algebra and seeing how it works, rather than just consulting tables such as Sloane's.Macbi wrote:As an aside, I think that [... quasars ...] can be extended in exactly the same way as lakes.
just as the first post on this thread said. The theory includes self-avoidance and closure, so it is mostly a question of understanding and applying it. What I hadn't previously realized is the multitude of extensible Life artifacts to which it is applicable, provoked by an encounter with "Beehive at beehive" and "Beehive at loaf." Of course there is also a "Loaf at loaf" and both can be convoluted almost at will. Somehow it seems that Life's vocabulary takes some of the blame for all the proliferation - specifically the separation between strict still life and "pseudo" still life.I wrote:All of this seems to be an exercise in random walks
I wrote:Could we use some better definitions here and there?" Also, are things like "tail" and "claw" defined, or are they just informal terms?
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