Permutations of the form a_1<a_2>a_3< >a_n

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The discussion centers on counting permutations of the set {1, 2, 3, ..., n} that follow the specific pattern a_1 < a_2 > a_3 < ... > a_n. The initial analysis reveals that there are (n-1) options for the first element, with restrictions based on whether the choice is odd or even. For odd choices, selecting the lowest element limits the next choice, while for even choices, selecting the highest element imposes similar restrictions. The complexity of the problem suggests that a different analytical approach may be necessary for effective counting.

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joeblow
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I've been thinking about this one for over a week now. Does anyone have any smart way of counting the permutations of {1,2,3,...,n} that are of the form a_1< a_2 > a_3 < ... > a_n?

You notice that there are (n-1) ways to choose the first elt. since the first choice cannot be n. And the complicated thing is that for an odd choice, if you choose the lowest possible elt., then the next choice cannot be the lowest poss elt. For even choices, if you choose the highest poss. elt., you cannot choose the highest elt. for the next choice. So, clearly thinking about this in terms of how many ways to choose individual elts. is probably not optimal. Can someone suggest another way of thinking about it? Thanks.
 
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