Spanning Trees in Grid Graphs

Paul Raff
Rutgers University
praff@math.rutgers.edu

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

$\left(\begin{array}{ccccccccccccccc}45& 24& 30& 24& 24& 36& 24& 30& 11& 12& 11& 11& 11& 12& 4\\ 9& 8& 6& 4& 5& 8& 4& 6& 3& 4& 3& 2& 2& 2& 1\\ 6& 4& 9& 4& 4& 4& 4& 4& 3& 2& 2& 3& 2& 2& 1\\ 9& 4& 6& 8& 4& 8& 5& 6& 3& 2& 2& 2& 3& 4& 1\\ 9& 5& 6& 4& 8& 8& 4& 6& 2& 4& 2& 3& 3& 2& 1\\ 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 9& 4& 6& 5& 4& 8& 8& 6& 2& 2& 3& 3& 2& 4& 1\\ 6& 4& 4& 4& 4& 4& 4& 9& 2& 2& 3& 2& 3& 2& 1\\ 5& 4& 6& 4& 3& 4& 3& 4& 3& 2& 2& 2& 2& 2& 1\\ 1& 1& 0& 0& 1& 2& 0& 0& 0& 1& 0& 0& 0& 0& 0\\ 5& 4& 4& 3& 3& 4& 4& 6& 2& 2& 3& 2& 2& 2& 1\\ 5& 3& 6& 3& 4& 4& 4& 4& 2& 2& 2& 3& 2& 2& 1\\ 5& 3& 4& 4& 4& 4& 3& 6& 2& 2& 2& 2& 3& 2& 1\\ 1& 0& 0& 1& 0& 2& 1& 0& 0& 0& 0& 0& 0& 1& 0\\ 4& 3& 4& 3& 3& 4& 3& 4& 2& 2& 2& 2& 2& 2& 1\end{array}\right)$

$\left(\text{⁠}\begin{array}{c}4\\ 1\\ 1\\ 1\\ 1\\ 0\\ 1\\ 1\\ 1\\ 0\\ 1\\ 1\\ 1\\ 0\\ 1\end{array}\text{⁠}\right)$

$-x^{15}+111x^{14}-2797x^{13}+33155x^{12}-223361x^{11}+919439x^{10}-2370069x^9+3823867x^8-3823867x^7+2370069x^6-919439x^5+223361x^4-33155x^3+2797x^2-111x+1$

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

$-\left(x-1\right){\left(x^2-6x+1\right)}^3{\left(x^2-4x+1\right)}^2\left(x^4-84x^3+230x^2-84x+1\right)$

$1-90x+735x^2-1548x^3+735x^4-90x^5+x^6$

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

$\frac{4x\left(x^4+6x^3-30x^2+6x+1\right)}{x^6-90x^5+735x^4-1548x^3+735x^2-90x+1}$

Created by Wolfram Mathematica 6.0  (06 March 2009)