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

In summary, the conversation discusses a tricky problem of counting permutations of a set that follow a specific pattern. The participants explore different approaches, including choosing individual elements and finding a specific page on Google that may offer a solution.
  • #1
joeblow
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0
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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  • #2

1. What are permutations of the form a_1a_3< >a_n?

Permutations of the form a_1a_3< >a_n are a type of mathematical arrangement where a set of elements are arranged in a specific order, with some elements separated by a symbol (such as < or >) to indicate their relative positions.

2. How do you calculate the number of permutations of this form?

The number of permutations of the form a_1a_3< >a_n can be calculated using the formula n!/(m!(n-m)!), where n is the total number of elements in the set and m is the number of elements that are separated by a symbol.

3. Can the elements in this form be repeated?

Yes, the elements in this form can be repeated. However, the number of permutations will be different depending on whether repetition is allowed or not.

4. How is this type of permutation used in real life?

Permutations of the form a_1a_3< >a_n can be used to analyze and solve problems in various fields such as computer science, statistics, and genetics. For example, in genetics, this type of permutation can be used to study the possible combinations of genes in a population.

5. Are there any special properties of permutations of this form?

Yes, permutations of this form have some special properties such as being reflexive, symmetric, and transitive. They can also be used to model various real-life scenarios, such as arranging objects in a specific order or determining the possible outcomes of a sequence of events.

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