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hw8 - Math 146 Algebraic Combinatorics Homework 8 This...

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Math 146: Algebraic Combinatorics Homework 8 This problem set is due Friday, May 21. This is a half problem set. Do the following problems: GK8.1. An alternating permutation is a permutation σ of [ n ] such that σ ( k + 1 ) - σ ( k ) is positive when k is odd and negative when k is even. For example, 1423 is an alternating permutation. Let a n be the number of alternating permutations of [ n ] , and let A ( x ) be their exponential generating function. (a) Let b n be the set of “zigzag permutations”, which are those that are either alternating or satisfy the reverse condition, that σ ( k + 1 ) - σ ( k ) is positive when k is even and negative when k is odd. Show that b n = 2 a n and thus B ( x ) = 2 A ( x ) for their generating functions. (We set b 0 = 2 by default.) *(b) Prove the “two animal” relation B 0 ( x ) = A ( x ) 2 + 1 . (c) Use (a) and (b) and the initial condition A ( 0 ) = a 0 = 1 to establish that A ( x ) = ( sec x )+( tan x ) . GK8.2. The version of Abel’s theorem that I gave in class, although correct, is not the strongest one avail- able for our purposes. It is a theorem that if f ( x )
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