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Unformatted text preview: 1. Determine the rook polynomial of the following board: In how many ways can you place 6 nontaking rooks on this board? In how many ways can you place them on the complementary board? The rook polynomial is equal to r ( C ) = r r = r r + x · r = (1 + 6 x + 6 x 2 ) bracketleftBig (1 + 2 x )(1 + 3 x + x 2 ) + x (1 + x )(1 + 2 x ) bracketrightBig = 1 + 12 x + 52 x 2 + 100 x 3 + 84 x 4 + 24 x 5 Since the coefficient at x 6 is zero, there is no way how to place 6 non taking rooks on this board. On the complementary board, there are 1 · 6! − 12 ·...
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 Spring '09
 MarniMishna
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