Spanning Trees in Grid Graphs

Paul Raff
Rutgers University
praff@math.rutgers.edu

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

$\left(\begin{array}{ccccccccccccccc}20& 12& 12& 8& 8& 12& 8& 12& 4& 4& 4& 3& 3& 3& 1\\ 1& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 1& 1& 3& 1& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0\\ 9& 5& 8& 8& 4& 6& 4& 6& 4& 2& 2& 2& 3& 3& 1\\ 9& 5& 6& 4& 8& 8& 4& 6& 2& 4& 2& 3& 3& 2& 1\\ 1& 1& 0& 0& 1& 3& 0& 0& 0& 1& 0& 0& 0& 0& 0\\ 9& 5& 6& 4& 4& 6& 8& 8& 2& 2& 4& 3& 2& 3& 1\\ 1& 1& 0& 0& 0& 0& 1& 3& 0& 0& 1& 0& 0& 0& 0\\ 1& 1& 2& 1& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0\\ 1& 1& 0& 0& 1& 2& 0& 0& 0& 1& 0& 0& 0& 0& 0\\ 1& 1& 0& 0& 0& 0& 1& 2& 0& 0& 1& 0& 0& 0& 0\\ 5& 3& 6& 3& 4& 4& 4& 4& 2& 2& 2& 3& 2& 2& 1\\ 5& 3& 4& 4& 4& 4& 3& 6& 2& 2& 2& 2& 3& 2& 1\\ 5& 3& 4& 4& 3& 6& 4& 4& 2& 2& 2& 2& 2& 3& 1\\ 4& 3& 4& 3& 3& 4& 3& 4& 2& 2& 2& 2& 2& 2& 1\end{array}\right)$

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

$-x^{15}+67x^{14}-1387x^{13}+14137x^{12}-82992x^{11}+301152x^{10}-700262x^9+1063202x^8-1063202x^7+700262x^6-301152x^5+82992x^4-14137x^3+1387x^2-67x+1$

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

$-\left(x-1\right){\left(x^2-6x+1\right)}^3{\left(x^2-3x+1\right)}^2\left(x^4-42x^3+83x^2-42x+1\right)$

$1-48x+336x^2-582x^3+336x^4-48x^5+x^6$

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

$\frac{x\left(x^4+6x^3-16x^2+6x+1\right)}{x^6-48x^5+336x^4-582x^3+336x^2-48x+1}$

Created by Wolfram Mathematica 6.0  (06 March 2009)