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4204finalproject - n vertices There is a completely...

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Combinatorics (MAD 4203) — Spring 2011 Due Wednesday 4/27 Take-home Final Project 1 Prove that it is possible to color each point of the plane either red or blue so that there is no equilateral triangle with side lengths one which has monochro- matic vertices. 2 Let P be a poset on n elements. Prove that P contains either a chain or an antichain with at least n elements. Then use this to prove the Erd˝os- Szekeres Theorem, which states that every permutation of length n contains an increasing or decreasing subsequence of length at least n . 3 In class, we used the M¨obius function of the set partition lattice to obtain an expression for the number of labeled, connected graphs on
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Unformatted text preview: n vertices. There is a completely diFerent approach to this enumeration problem. Use the Expo-nential ±ormula to compute the exponential generating function for labeled, connected graphs. 4 Give a probabilistic proof that as n tends to in²nity, the proportion of labeled graphs on n vertices which are connected tends to 1. 5 The inversion graph of the permutation π of length n , denoted G π , is the graph on [ n ] where i ∼ j if and only if i < j and π ( i ) > π ( j ). ±or which permutations π is G π a path? 6 Prove that the growth rate of Av( k ··· 21) is ( k-1) 2 . (We will discuss growth rates on Monday, 4/11.)...
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