Spanning Trees in Grid Graphs

Paul Raff
Rutgers University
praff@math.rutgers.edu

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

$\left(\begin{array}{ccccccccccccccc}21& 11& 14& 11& 8& 15& 8& 9& 5& 4& 3& 3& 3& 4& 1\\ 3& 3& 2& 1& 1& 2& 0& 0& 1& 1& 0& 0& 0& 0& 0\\ 1& 1& 3& 1& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0\\ 3& 1& 2& 3& 0& 2& 1& 0& 1& 0& 0& 0& 0& 1& 0\\ 9& 5& 6& 4& 8& 8& 4& 6& 2& 4& 2& 3& 3& 2& 1\\ 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 9& 4& 6& 5& 4& 8& 8& 6& 2& 2& 3& 3& 2& 4& 1\\ 6& 4& 4& 4& 4& 4& 4& 9& 2& 2& 3& 2& 3& 2& 1\\ 1& 1& 2& 1& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0\\ 1& 1& 0& 0& 1& 2& 0& 0& 0& 1& 0& 0& 0& 0& 0\\ 5& 4& 4& 3& 3& 4& 4& 6& 2& 2& 3& 2& 2& 2& 1\\ 5& 3& 6& 3& 4& 4& 4& 4& 2& 2& 2& 3& 2& 2& 1\\ 5& 3& 4& 4& 4& 4& 3& 6& 2& 2& 2& 2& 3& 2& 1\\ 1& 0& 0& 1& 0& 2& 1& 0& 0& 0& 0& 0& 0& 1& 0\\ 4& 3& 4& 3& 3& 4& 3& 4& 2& 2& 2& 2& 2& 2& 1\end{array}\right)$

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

$-x^{15}+69x^{14}-1461x^{13}+14913x^{12}-86655x^{11}+310379x^{10}-714267x^9+1078719x^8-1078719x^7+714267x^6-310379x^5+86655x^4-14913x^3+1461x^2-69x+1$

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

$-\left(x-1\right)\left(x^2-4x+1\right)\left(x^4-8x^3+16x^2-8x+1\right)\left(x^8-56x^7+672x^6-2632x^5+4094x^4-2632x^3+672x^2-56x+1\right)$

$1-56x+672x^2-2632x^3+4094x^4-2632x^5+672x^6-56x^7+x^8$

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

$\frac{x\left(x^6-49x^4+112x^3-49x^2+1\right)}{x^8-56x^7+672x^6-2632x^5+4094x^4-2632x^3+672x^2-56x+1}$

Created by Wolfram Mathematica 6.0  (06 March 2009)