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August 29th, 2014, 01:49 PM  #1 
Newbie Joined: Aug 2014 From: Spain Posts: 4 Thanks: 0  How many ways are there to color the Hshaped tree with 3 colors such that each c How many ways are there to color this graph with the following constraints? We have three colors: blue, red, green, and we require that the number of nodes of color green is 2, and blue 2, and red 2  the same number. And here is my attempt: Automorphisms: $$(1)(2)(3)(4)(5)(6)$$ $$(34)(1)(2)(5)(6)$$ $$(56)(1)(2)(3)(4) $$ $$(34)(56)(1)(2)$$ $$(12)(35)(46)$$ $$(12)(36)(45)$$ Cycle index of group: $$Z_G(x_1,...,x_6) =\frac16 (x_1^6 + 2x_2^3 + 2x_2x_1^4 + x_2^2x_1^2) $$ And using PĆ³lya theorem we get generating function: $$U_D(g,r,b) = Z_G(g+r+b, g^2+r^2+b^2, ...,g^6+r^6+b^6) = \\ \frac16 ((g+r+b)^6 + 2(g^2+r^2+b^2)^3 + 2(g^2+r^2+b^2)(g+r+b)^4 + (g^2+r^2+b^2)^2(g+r+b)^2)$$ And coefficient with $r^2g^2b^2$ is $\frac16 (90 + 12 +0 + 0 ) = 17$ Is it solution correct? 
August 29th, 2014, 02:18 PM  #2 
Senior Member Joined: Sep 2012 From: British Columbia, Canada Posts: 764 Thanks: 53 
You might want to post this in the "Abstract Algebra" section in the college math forum rather than in the prealgebra section of elementary math, so it attracts more helpful attention.

August 29th, 2014, 08:11 PM  #3 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,898 Thanks: 1093 Math Focus: Elementary mathematics and beyond 
Moved to Abstract Algebra.


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$h$shaped, color, colors, hshaped, tree, ways 
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