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>*Reply-to*: cfo at lanl.gov

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 -- ========================================================== Curtis Osterhoudt cfo at remove_this.lanl.and_this.gov PGP Key ID: 0x4DCA2A10 Please avoid sending me Word or PowerPoint attachments See http://www.gnu.org/philosophy/no-word-attachments.html ==========================================================

**References**:**Inequalities or histogram or something.......***From:*Steve Gray <stevebg@roadrunner.com>