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# 4203hw3 - concerns Chapter 3 5 Without using any asymptotic...

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Combinatorics (MAD 4203) — Fall 2010 Due Friday 10/8/2010 Homework #3 1 Find the number of compositions of 25 into ±ve odd parts. 2 Let a n denote the number of compositions of n into parts which are all greater than 1. Express a n in terms of a n - 1 and a n - 2 . 3 Let b n denote the number of compositions of n into parts which are all greater than 2. Find a recurrence for b n similar to the recurrence you found in the previous exercise. 4 Suppose that you have 4 white rooks and 4 black rooks. In how many ways can you place these on an 8 × 8 chessboard so that no two rooks can attack each other? 1 Note that two con±gurations are considered to be di²erent if they either have rooks in di²erent positions, or di²er- ent colored rooks in the same positions. Also note that this problem
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Unformatted text preview: concerns Chapter 3. 5 Without using any asymptotic bounds for p ( n ), prove that p ( n ) grows faster than any polynomial. That is, if f ( n ) is any polynomial, prove that there is some integer N such that p ( n ) > f ( n ) for all n > N . 6 Suppose that 1 ≤ k < n . Prove that the part k occurs a total of ( n-k + 3)2 n-k-2 times among all the 2 n-1 compositions of n . For example, if n = 4 and k = 2, then the part 2 occurs once in 2 + 1 + 1, 1 + 2 + 1, and 1 + 1 + 2, and twice in 2 + 2, for a total of 5 = (4-2 + 3)2 4-2-2 times. 1 Rooks can attack pieces in their same row or column only....
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