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hw7 - isomorphic if there is a bijection f X → X such...

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HOMEWORK #7, DUE ON 12/06 (1) Suppose that each vertex of regular n -gon is labeled with either 0 or 1. For which values of n is the number of different labellings under the action of C n equal to the number of different labellings of under the action of D n ? (2) (Exercise 48 of Chapter 14) A stained glass window in the form of a 3 × 3 chessboard has nine squares, each of which is colored red or blue (the colors are transparent and the window can be looked at from either side). Determine the generating function for the number of different stained glass windows, and total number of stained glass windows. (3) Let X be a finite set. Two binary relations R 1 and R 2 are called
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Unformatted text preview: isomorphic , if there is a bijection f : X → X such that ( a,b ) ∈ R 1 i± ( f ( a ) ,f ( b )) ∈ R 2 . Count the binary relations of a 4-element set, if isomorphic relations are considered to be identical. (4) Let V be ²nite set. A graph on V is an irre³exive, symmetric relation on V . Count the graphs on a 5-elements set, if isomorphic graphs (de²ned by isomorphic relations) are considered to be identical. ( Hint: This is basically Exercise 54 of Chapter 14, and the method is outlined on pages 575–576. ) (5) Compute the number of non-equivalent colorings of the faces of a square pyramid with k colors. 1...
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