Spanning Trees in Grid Graphs

Paul Raff
Rutgers University
praff@math.rutgers.edu

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

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$x^{52}-313x^{51}+39241x^{50}-2803356x^{49}+132009608x^{48}-4435805394x^{47}+111791069937x^{46}-2186482208211x^{45}+34025561189388x^{44}-429364991056061x^{43}+4459652527751271x^{42}-38589748763072122x^{41}+280968944489755513x^{40}-1735684454457683189x^{39}+9161394166135977052x^{38}-41565342835535396119x^{37}+162932284766756125755x^{36}-554242960654742096726x^{35}+1642290223023744855093x^{34}-4252629019512404906991x^{33}+9649657853880590343636x^{32}-19231365441915041099265x^{31}+33726621686559464547923x^{30}-52126281677266845332958x^{29}+71082666288810739168099x^{28}-85594843403689196715295x^{27}+91058174715381930686966x^{26}-85594843403689196715295x^{25}+71082666288810739168099x^{24}-52126281677266845332958x^{23}+33726621686559464547923x^{22}-19231365441915041099265x^{21}+9649657853880590343636x^{20}-4252629019512404906991x^{19}+1642290223023744855093x^{18}-554242960654742096726x^{17}+162932284766756125755x^{16}-41565342835535396119x^{15}+9161394166135977052x^{14}-1735684454457683189x^{13}+280968944489755513x^{12}-38589748763072122x^{11}+4459652527751271x^{10}-429364991056061x^9+34025561189388x^8-2186482208211x^7+111791069937x^6-4435805394x^5+132009608x^4-2803356x^3+39241x^2-313x+1$

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

${\left(x-1\right)}^4\left(x^2-3x+1\right)\left(x^6-13x^5+56x^4-93x^3+56x^2-13x+1\right)\left(x^{12}-53x^{11}+991x^{10}-8540x^9+37243x^8-88091x^7+117046x^6-88091x^5+37243x^4-8540x^3+991x^2-53x+1\right)\left(x^{12}-39x^{11}+561x^{10}-3858x^9+14231x^8-30327x^7+38866x^6-30327x^5+14231x^4-3858x^3+561x^2-39x+1\right)\left(x^{16}-201x^{15}+11104x^{14}-259893x^{13}+3001225x^{12}-18824856x^{11}+67848270x^{10}-144802410x^9+186068896x^8-144802410x^7+67848270x^6-18824856x^5+3001225x^4-259893x^3+11104x^2-201x+1\right)$

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

$-\left(x\left(x^{14}-1425x^{12}+26532x^{11}-180448x^{10}+545916x^9-661242x^8+661242x^6-545916x^5+180448x^4-26532x^3+1425x^2-1\right)\right)/\left(x^{16}-201x^{15}+11104x^{14}-259893x^{13}+3001225x^{12}-18824856x^{11}+67848270x^{10}-144802410x^9+186068896x^8-144802410x^7+67848270x^6-18824856x^5+3001225x^4-259893x^3+11104x^2-201x+1\right)$

Created with Wolfram Mathematica 7.0