Two ordering problems

[franktaw at netscape.net]

Apparently you missed the discussion between Max and myself.  The
answer to your first problem (as defined below) is A005806.

The ones that can be achieved by assigning primes are A009997.  Since
this is different from A005806, the answer to the second problem is no.

Franklin T. Adams-Watters

-----Original Message-----
From: David Wilson <dwilson at gambitcomm.com>

If we momentarily ignore the problem of finding numerical examples for 
a
moment, let P(n) be a set of n distinct symbolic primes (e.g, P(3) =
{p,q,r}) and let D(n) be the set of divisors of PROD(P(n)) (e.g, D(3) =
{1,p,q,r,pq,qr,pr,pqr}. We then use the following rules 

   [1] 1 < p < q < r < ... < x 
   [2] a < b  =>  ac < bc 
   [3] (a < b and b < c) => a < c 

to induce a partial order G(n) on D(n) (G(n) because it can be
represented as a graph with nodes D(n)). 

The first problem is to count the total orderings of D(n) that conform
to G(n). I doubt we are the first to consider this problem, I wonder if
there is an efficient algorithm. 

The second problem is whether all of the total orderings can be 
realized
by assignment of numerical primes to the variables. Interesting 
question. 

...





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