Re: Inequalities or histogram or something.......

• To: mathgroup at smc.vnet.net
• Subject: [mg89359] Re: [mg89338] Inequalities or histogram or something.......
• From: Curtis Osterhoudt <cfo at lanl.gov>
• Date: Sat, 7 Jun 2008 02:57:36 -0400 (EDT)
• Organization: LANL
• References: <200806061047.GAA24209@smc.vnet.net>

```Hi, Steve,

What follows is NOT a symbolic solution, but a numeric approach. I'll keep
working on finding a symbolic solution (I love this kind of thing), but
probably someone from the list has already beaten me to it.

(* Create a list of solutions up v = 50. If the upper limit on v is changed,
the number of solutions sought in FindInstance may need to be changed, too*)
solns = Table[{k, v,
Evaluate[
FindInstance[
k == x1 + x3 - x2 - 2 &&
Inequality[1, LessEqual, x1, Less, x2,
Less, x3, LessEqual, v], {x1, x2, x3}, Integers,
1000]]}, {v, 3, 50}, {k, 1, v - 1}];

(* Curry the solutions to pick out the Lengths of the solution sets *)
nums1 = solns /. {a_, b_, c_} :> {a, b, Length[c]};

(* Further curry the result to extract non-anomalous results. I don't know why
my "nums1" list sometimes has non-numeric values for the lengths*)
nums2 = Select[Flatten[nums1, 1], NumberQ[#1[[3]]] & ];

(* Plot the result as a tabulation vs. k and v *)
ListPlot3D[nums2, PlotRange -> All, InterpolationOrder -> 0,
AxesLabel -> {k, v}]

On Friday 06 June 2008 04:47:03 Steve Gray wrote:
> 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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```

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