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Zig and Zag

An alternating permutation is an arrangement of a set of numbers; specifically, the numbers are arranged so that the value of any particular number is not in the range of the values on either side of it. Mathematically:

Let A={c1,c2,...,cn} be an alternating permutation of the numbers c1,c2,...,cn. Then neither c(i-1)<c(i)<c(i+1) nor c(i-1)>c(i)>c(i+1) for all naturals i.

So suppose A={1,2,3,4,5}. An alternating permutation of A is the set {1,5,2,4,3} (5 is not between 1 and 2, 2 is not between 5 and 4, and 4 is not between 2 and 3 in terms of the number value).

The number of alternating permutations for a specific number of naturals is called an Euler zigzag number (just how many different Euler numbers are there?) and is denoted A(n) (or An). The first several values of A(n) are as follows:

n  A(n)
0  1
1  1
2  1
3  2
4  5
5  16
6  61
7  272
.
.
.

An interesting property of the zigzag numbers (leading to the actual zig and zag distinctions) is the following trigonometric equation involving a Maclaurin series:

                  ∞
                -----
                \        /   n  \
sec(x)+tan(x) =  >    A |   x    |
                /      n| ------ |
                -----    \  n!  /
                 n=1

(Here the even-numbered terms [A(0), A(2), etc.] are associated with the secant, while the odd-numbered ones are associated with the tangent, see zig number and zag number respectively.)


Major source: http://mathworld.wolfram.com/AlternatingPermutation.html.

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