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Inequalities or histogram or something.......

  • To: mathgroup at smc.vnet.net
  • Subject: [mg89338] Inequalities or histogram or something.......
  • From: Steve Gray <stevebg at roadrunner.com>
  • Date: Fri, 6 Jun 2008 06:47:03 -0400 (EDT)

Can Mathematica help with this? Or can someone?

I have positive integers x1,x2,x3,k,v.

There are assumptions:
1 <= k < v and
1 <= x1 < x2 < x3 <= v.

There is one equation:
k = x1 + x3 - x2 - 2.

I need a symbolic solution for the number of combinations of x1,x2,x3
that satisfy the equation under the assumptions.

This will be a histogram of k vs. the number of solutions.
One numeric point on the histo: if v=8 and k=0, there are 6 solutions,
x1,x2,x3 = 1,2,3; 1,3,4; 1,4,5; 1,5,6; 1,6,7; 1,7,8; none with x1 > 1.

I need a general symbolic solution in terms of D(v,k). I need a way to
show its derivation for a paper. I happen to know that 
D(v,k) + D(v,v-k-3) = 2(k+1)(v-k-2).

This is not overwhelmingly complicated but I don't know a decent way
to go about it. Thank you for any help, using Mathematica or not. (I have
version 6.)

Steve Gray


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