Spanning Trees in Grid Graphs

Paul Raff
Rutgers University
praff@math.rutgers.edu

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

$\left(\begin{array}{ccccccccccccccc}75& 45& 60& 45& 40& 55& 40& 55& 24& 21& 21& 20& 21& 21& 8\\ 9& 8& 8& 5& 4& 6& 4& 6& 4& 3& 3& 2& 2& 2& 1\\ 1& 1& 3& 1& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0\\ 9& 5& 8& 8& 4& 6& 4& 6& 4& 2& 2& 2& 3& 3& 1\\ 19& 11& 16& 11& 16& 16& 11& 16& 6& 8& 6& 8& 8& 6& 3\\ 6& 4& 4& 4& 4& 9& 4& 4& 2& 3& 2& 2& 2& 3& 1\\ 19& 11& 16& 11& 11& 16& 16& 16& 6& 6& 8& 8& 6& 8& 3\\ 6& 4& 4& 4& 4& 4& 4& 9& 2& 2& 3& 2& 3& 2& 1\\ 1& 1& 2& 1& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0\\ 5& 4& 4& 3& 4& 6& 3& 4& 2& 3& 2& 2& 2& 2& 1\\ 5& 4& 4& 3& 3& 4& 4& 6& 2& 2& 3& 2& 2& 2& 1\\ 5& 3& 6& 3& 4& 4& 4& 4& 2& 2& 2& 3& 2& 2& 1\\ 5& 3& 4& 4& 4& 4& 3& 6& 2& 2& 2& 2& 3& 2& 1\\ 5& 3& 4& 4& 3& 6& 4& 4& 2& 2& 2& 2& 2& 3& 1\\ 4& 3& 4& 3& 3& 4& 3& 4& 2& 2& 2& 2& 2& 2& 1\end{array}\right)$

$\left(\text{⁠}\begin{array}{c}8\\ 1\\ 0\\ 1\\ 3\\ 1\\ 3\\ 1\\ 0\\ 1\\ 1\\ 1\\ 1\\ 1\\ 1\end{array}\text{⁠}\right)$

$-x^{15}+161x^{14}-4827x^{13}+65739x^{12}-498985x^{11}+2274617x^{10}-6355787x^9+10761739x^8-10761739x^7+6355787x^6-2274617x^5+498985x^4-65739x^3+4827x^2-161x+1$

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

$-\left(x-1\right){\left(x^2-6x+1\right)}^2{\left(x^2-4x+1\right)}^3\left(x^4-136x^3+1170x^2-136x+1\right)$

$1-140x+1715x^2-4952x^3+1715x^4-140x^5+x^6$

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

$\frac{8x\left(x^4+4x^3-70x^2+4x+1\right)}{x^6-140x^5+1715x^4-4952x^3+1715x^2-140x+1}$

Created by Wolfram Mathematica 6.0  (06 March 2009)