Spanning Trees in Grid Graphs

Paul Raff
Rutgers University
praff@math.rutgers.edu

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

$\left(\begin{array}{ccccccccccccccc}40& 24& 29& 21& 16& 24& 21& 29& 11& 8& 11& 8& 8& 9& 3\\ 3& 3& 2& 1& 0& 0& 1& 2& 1& 0& 1& 0& 0& 0& 0\\ 1& 1& 3& 1& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0\\ 9& 5& 8& 8& 4& 6& 4& 6& 4& 2& 2& 2& 3& 3& 1\\ 19& 11& 16& 11& 16& 16& 11& 16& 6& 8& 6& 8& 8& 6& 3\\ 6& 4& 4& 4& 4& 9& 4& 4& 2& 3& 2& 2& 2& 3& 1\\ 9& 5& 6& 4& 4& 6& 8& 8& 2& 2& 4& 3& 2& 3& 1\\ 1& 1& 0& 0& 0& 0& 1& 3& 0& 0& 1& 0& 0& 0& 0\\ 1& 1& 2& 1& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0\\ 5& 4& 4& 3& 4& 6& 3& 4& 2& 3& 2& 2& 2& 2& 1\\ 1& 1& 0& 0& 0& 0& 1& 2& 0& 0& 1& 0& 0& 0& 0\\ 5& 3& 6& 3& 4& 4& 4& 4& 2& 2& 2& 3& 2& 2& 1\\ 5& 3& 4& 4& 4& 4& 3& 6& 2& 2& 2& 2& 3& 2& 1\\ 5& 3& 4& 4& 3& 6& 4& 4& 2& 2& 2& 2& 2& 3& 1\\ 4& 3& 4& 3& 3& 4& 3& 4& 2& 2& 2& 2& 2& 2& 1\end{array}\right)$

$\left(\text{⁠}\begin{array}{c}3\\ 0\\ 0\\ 1\\ 3\\ 1\\ 1\\ 0\\ 0\\ 1\\ 0\\ 1\\ 1\\ 1\\ 1\end{array}\text{⁠}\right)$

$-x^{15}+105x^{14}-2743x^{13}+32929x^{12}-219362x^{11}+879502x^{10}-2200510x^9+3478928x^8-3478928x^7+2200510x^6-879502x^5+219362x^4-32929x^3+2743x^2-105x+1$

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

$-\left(x-1\right)\left(x^2-6x+1\right)\left(x^2-5x+1\right)\left(x^2-3x+1\right)\left(x^8-90x^7+1313x^6-5850x^5+9828x^4-5850x^3+1313x^2-90x+1\right)$

$1-90x+1313x^2-5850x^3+9828x^4-5850x^5+1313x^6-90x^7+x^8$

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

$\frac{3x\left(x^6-67x^4+180x^3-67x^2+1\right)}{x^8-90x^7+1313x^6-5850x^5+9828x^4-5850x^3+1313x^2-90x+1}$

Created by Wolfram Mathematica 6.0  (06 March 2009)