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Adv Alegbra HW Solutions 80

Adv Alegbra HW Solutions 80 - 80 Solution True 2.138 How...

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80 Solution. True. 2.138 How many fl ags are there with n stripes each of which can be colored any one of q given colors? Solution. We use Proposition 2.164. Here, G = τ , where τ = ( 1 n )( 2 n 1 ) · · · ( k k + 1 ) if n = 2 k , and τ = ( 1 n )( 2 n 1 ) · · · ( k 1 k + 1 )( k ) if n = 2 k + 1. Therefore, P G ( x 1 , . . . , x n ) = 1 2 ( x n 1 + x k 2 ) if n = 2 k 1 2 ( x n 1 + x 1 x k 2 ) if n = 2 k + 1 . Thus, P G ( q , . . . , q ) = 1 2 ( q n + q k ) or 1 2 ( q n + q k + 1 ) , depending on the parity of n . One can give a master formula: P G ( q , . . . , q ) = 1 2 ( q n + q [ n / 2 ] ). 2.139 Let X be the squares in an n × n grid, and let ρ be a rotation by 90 . De fi ne a chessboard to be a ( q , G ) -coloring, where the cyclic group G = ρ of order 4 is acting. Show that the number of chessboards is 1 4 q n 2 + q ( n 2 + 1 )/ 2 + 2 q ( n 2 + 3 )/ 4 , where x is the greatest integer in the number x . Solution. The group G that acts here is a cyclic group ρ of order 4, where ρ is (clockwise) rotation by 90 . As n 0 , 1 , 2 , or 3 mod 4 , we have n 2 0 or 1 mod 4. Now
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