Spanning Trees in Grid Graphs

Paul Raff
Rutgers University
praff@math.rutgers.edu

$\mathrm{ShowGraph}\left[\mathrm{GraphProduct}\left[\mathrm{PathGraph}\left[5\right],\mathrm{ToMathematicaGraph}\left[G,3\right]\right]\right]$

$\left(\begin{array}{ccccc}16& 8& 8& 8& 3\\ 4& 3& 2& 2& 1\\ 4& 2& 3& 2& 1\\ 4& 2& 2& 3& 1\\ 3& 2& 2& 2& 1\end{array}\right)$

$\left(\text{⁠}\begin{array}{c}3\\ 1\\ 1\\ 1\\ 1\end{array}\text{⁠}\right)$

$-x^5+26x^4-73x^3+73x^2-26x+1$

$\mathrm{Factor}\left[\mathrm{char}\right]//\mathrm{TraditionalForm}$

$-{\left(x-1\right)}^3\left(x^2-23x+1\right)$

$-1+24x-24x^2+x^3$

$\mathrm{GeneratingFunctionFromPolynomialAndSequence}\left[\mathrm{rec},\mathrm{seq}\right]//\mathrm{TraditionalForm}$

$-\frac{3x\left(x+1\right)}{x^3-24x^2+24x-1}$

Created by Wolfram Mathematica 6.0  (06 March 2009)