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I'm really bad with Combinatorics and I can't find similar problems to this one online either. To me, the question isn't very specific which makes it harder for me to find the answer.

A math teacher ordered 13 girls in a class to form a line such that no girl would have girls shorter than her on both sides. All girls in the class are of different height. In how many ways can the girls carry out the teacher's commands?

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    Doesn't the requirement implies that the line must be either all decreasing or increasing? – 2017-01-08
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    A line or a circle? If a line, must the end girls be shorter than their only neighbor? – 2017-01-08
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    Assuming you did mean a line and that there is no condition on the end girls, Hint: where can the tallest girl be? once you've placed here, where can the second tallest be? And then the third tallest, and so on. – 2017-01-08
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    It should be a line... but the condition is that a tall girl cannot be beside two girls who are shorter than her, so it can be a short girl or tall girl at the end of the line. – 2017-01-08
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    The question does not specify if the line should be decreasing or increasing ... – 2017-01-08
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    Yes, but it's an immediate implication of the conditions. Try the hint I suggested. As a further suggestion, work the problem for a smaller group of girls. Start with three girls, then do four. You'll see the pattern. – 2017-01-08

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$\underline {HINT}$

Don't look at the tallest person, fix the shortest person at any of the available positions, and go on placing taller person(s) on adjacent side(s) one by one.
One such possibility for a group of $5$ is depicted pictorially below.

${\Huge...}\quad$Left end
${\Huge ..}$
${\Huge .}\quad$ Shortest person
${\Huge ....}$
${\Huge .....}\quad$Right end

FURTHER HINT

Count the number of possibilities with the shortest person at this position. Do such counts with the shortest person at each available position. Then add up.