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184a_99f_2e

# 184a_99f_2e - (c Let w n be the number of webs with vertex...

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Math 184A Second Exam 1 December 1999 1. (30 pts) (a) Find the strictly decreasing function f :4 20 of rank 50. (b) Determine the lex order rank of the strictly decreasing function 7 , 4 , 2 , 1. Show your work. 2. (15 pts) Find the 6-leaf binary (unlabeled rooted plane) tree whose rank is 20. Show your work. 3. (55 pts) Deﬁne a web recursively to be either (i) The simple graph with V =3 = { 1 , 2 , 3 } and E = n { 1 , 2 } , { 1 , 3 } , { 2 , 3 } o (a “triangle”) or (ii) A simple graph with V = n for some n> 3 such that vertex n has degree 2 and removing n and the two edges joining it gives a web with n - 1 vertices. Do the following. (a) Draw the 3 webs that have 4 vertices. (b) Prove that a web is not a tree.
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Unformatted text preview: (c) Let w n be the number of webs with vertex set n . Write down a recursion for w n and explain how you got it. (d) Prove that, for n ≥ 3 the number of webs with vertex set n is ( n-1)! ( n-2)! 2 n-2 . Here are some values of b n and binomial coeﬃcients. b 1 = 1 b 2 = 1 b 3 = 2 b 4 = 5 b 5 = 14 b 6 = 42 . ± 6 4 ¶ = 15 ± 7 4 ¶ = 35 ± 8 4 ¶ = 91 ± 5 3 ¶ = 10 ± 6 3 ¶ = 20 ± 7 3 ¶ = 35 ± 8 3 ¶ = 56 ± 4 2 ¶ = 6 ± 5 2 ¶ = 10 ± 6 2 ¶ = 15 ± 7 2 ¶ = 21 ± 8 2 ¶ = 28 ± 9 2 ¶ = 36 ± 10 2 ¶ = 45...
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