Spanning Trees in Grid Graphs

Paul Raff
Rutgers University
praff@math.rutgers.edu

$\mathrm{ShowGraph}\left[\mathrm{GraphProduct}\left[\mathrm{PathGraph}\left[5\right],\mathrm{ToMathematicaGraph}\left[G,4\right]\right]\right]$

$\left(\begin{array}{ccccccccccccccc}125& 75& 100& 75& 75& 100& 75& 100& 40& 40& 40& 40& 40& 40& 16\\ 19& 16& 16& 11& 11& 16& 11& 16& 8& 8& 8& 6& 6& 6& 3\\ 6& 4& 9& 4& 4& 4& 4& 4& 3& 2& 2& 3& 2& 2& 1\\ 19& 11& 16& 16& 11& 16& 11& 16& 8& 6& 6& 6& 8& 8& 3\\ 19& 11& 16& 11& 16& 16& 11& 16& 6& 8& 6& 8& 8& 6& 3\\ 6& 4& 4& 4& 4& 9& 4& 4& 2& 3& 2& 2& 2& 3& 1\\ 19& 11& 16& 11& 11& 16& 16& 16& 6& 6& 8& 8& 6& 8& 3\\ 6& 4& 4& 4& 4& 4& 4& 9& 2& 2& 3& 2& 3& 2& 1\\ 5& 4& 6& 4& 3& 4& 3& 4& 3& 2& 2& 2& 2& 2& 1\\ 5& 4& 4& 3& 4& 6& 3& 4& 2& 3& 2& 2& 2& 2& 1\\ 5& 4& 4& 3& 3& 4& 4& 6& 2& 2& 3& 2& 2& 2& 1\\ 5& 3& 6& 3& 4& 4& 4& 4& 2& 2& 2& 3& 2& 2& 1\\ 5& 3& 4& 4& 4& 4& 3& 6& 2& 2& 2& 2& 3& 2& 1\\ 5& 3& 4& 4& 3& 6& 4& 4& 2& 2& 2& 2& 2& 3& 1\\ 4& 3& 4& 3& 3& 4& 3& 4& 2& 2& 2& 2& 2& 2& 1\end{array}\right)$

$\left(\text{⁠}\begin{array}{c}16\\ 3\\ 1\\ 3\\ 3\\ 1\\ 3\\ 1\\ 1\\ 1\\ 1\\ 1\\ 1\\ 1\\ 1\end{array}\text{⁠}\right)$

$-x^{15}+235x^{14}-7909x^{13}+120319x^{12}-1025805x^{11}+5257407x^{10}-16329905x^9+29687219x^8-29687219x^7+16329905x^6-5257407x^5+1025805x^4-120319x^3+7909x^2-235x+1$

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

$-\left(x-1\right)\left(x^2-198x+1\right){\left(x^2-6x+1\right)}^6$

$1-204x+1190x^2-204x^3+x^4$

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

$\frac{16x\left(x^2+12x+1\right)}{x^4-204x^3+1190x^2-204x+1}$

Created by Wolfram Mathematica 6.0  (06 March 2009)