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# Monotonic Subsequences

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Posted 04 December 2013 - 02:20 AM

Consider a finite sequence of distinct integers. A subsequence is a sequence formed by deleting some items from the original sequence without disturbing their relative ordering. A subsequence is called monotone if it is either increasing (each term is larger than the one before it) or decreasing (each term is smaller than the one before it). For example, if the sequence is 4, 6, 3, 5, 7, 1, 2, 9, 8, 10, then 4, 6, 8, 10 is a monotone (increasing) subsequence of length 4 and 6, 5, 2 is a monotone (decreasing) subsequence of length 3.

a) Find a sequence of 9 distinct integers that has no monotone subsequence of length 4.

b) Show that every such sequence of length 10 has a monotone subsequence of length 4.

c) Generalize. How long must the sequence be to guarantee a monotone subsequence of length n?

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### #2 bonanova

bonanova

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Posted 05 December 2013 - 09:19 AM   Best Answer

Spoiler for starters

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The greatest challenge to any thinker is stating the problem in a way that will allow a solution.
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