Spanning Trees in Grid Graphs

Paul Raff
Rutgers University
praff@math.rutgers.edu

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

$\left(\begin{array}{ccccc}8& 4& 3& 3& 1\\ 1& 1& 0& 0& 0\\ 4& 2& 3& 2& 1\\ 4& 2& 2& 3& 1\\ 3& 2& 2& 2& 1\end{array}\right)$

$\left(\text{⁠}\begin{array}{c}1\\ 0\\ 1\\ 1\\ 1\end{array}\text{⁠}\right)$

$-x^5+16x^4-47x^3+47x^2-16x+1$

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

$-\left(x-1\right)\left(x^4-15x^3+32x^2-15x+1\right)$

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

$-\frac{x\left(x^2-1\right)}{x^4-15x^3+32x^2-15x+1}$

Created by Wolfram Mathematica 6.0  (06 March 2009)