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R is very good statistical software available free from www.r-project.org It does a lot of complicated stuff, but just connecting dots is easy to show in a demo: x1 = c(1,2,3,4,5,6,7) y1 = c(5.4, 5.7, 6.2, 6.9, 6.5, 5.8, 4.1) # 'plot' sets up axes, labels, etc. plot(x1, y1, type="l", col="blue", lty = "dotted", # 'ell' not 'one' ylim=c(0, 8), ...

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For plotting points, especially points in 3D space, try Graphing Calculator 3D: http://www.runiter.com/graphing-calculator/

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One option is gnuplot, which handles a lot of graph formats. More flexible in format (and easier to integrate with LaTeX) is asymptote, but then you have to use a library, and defining the graph is more work.

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Most office application suites have some tool to list data and draw plots accordingly. They also allow you to import your data easily. A free example is Libre Office, with its "Calc" application. The commercial equivalent would be Microsoft Office with Excel. Another type of program that can do this are Matlab, Mathematica, MathCAD, and the like, they ...

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As I understand you want certain vertices to leave trails as they change their position. You can do it by right-clicking on the vertex and checking the "Trace On" option. Now the vertex will leave traces of it's colour on the canvas.

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You could do this with a constraint solver, such as Savile Row ( http://savilerow.cs.st-andrews.ac.uk/ ). While this tool does not support permutations, and permutation multiplication, directly, you can map the problem to integers fairly easily. This wouldn't use any group theory techniques at all, just tackle the problem as permutations with search, but ...

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I am actually going to back up the answer (in slightly more generality) given by Robert Israel, and address the comments you made there. I would guess that the number of equations is not a big problem, the difficulty will be more focused on the number of variables (involved in each equation) and the size of the group. The reason why it is not a big deal ...

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I use Python and the NetworkX package: https://networkx.github.io/ Granted, there's some learning to be done in using it, but it's very straight forward and richly featured. And of course using Python makes the automation part simple too. Take a look. I've used it in a board game setting (machine learning) and finance. From their website: Features ...

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There are just $6^6$ possible assignments of permutations to variables to check, so a brute-force solution is probably the simplest in this case. In general, you want to enumerate the possible solutions, using the equations as early as possible to cut down on the branching. The more equations the better! Thus in this case you might have pseudocode something ...

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I have good experiences with LINGO from LINDO Systems. A free version (6 months) can be downloaded here. $\color{blue}{\texttt{Input example}}$ $MAX \ \ 5 X1 + 7 X2 \\ ST \\ 3 X1 + 4 X2 < 650\\ 2 X1 + 3 X2 < 500$ The default setting for the variables is, that they are non-negative. $X1, \ X2 \geq 0$ $\color{blue}{\texttt{Output ... 0 There are many linear programming solvers available. Look at Wikipedia, for example. I would think that nearly all of these would accept equations as well as inequalities. But if by some chance you're stuck with a program that only accepts inequalities, you can replace each equation$a = b$by the pair of inequalities$a \le b$and$a \ge b$0 You can actually do this really easily in Magma using the ColumnSubmatrix command, no looping necessary. You can use this in a few ways. For example if you have a matrix$A$and you want$B$to be made up a selection of columns: 1st, 2nd,$\ldots$, 5th columns: B := ColumnSubmatrix(A, 5); 3rd, 4th,$\ldots\$, 7th columns: B := ColumnSubmatrix(A, 3, ...

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Cytoscape is a pretty good tool for constructing and visualizing graphs. It provides you quite a few node and network parameters.

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