Alternating permutation

Alternating permutation

In combinatorial mathematics, an alternating permutation of the set {1, 2, 3, ..., "n"} is an arrangement of those numbers into an order "c"1, ..., "c""n" such that no element "c""i" is between "c""i" − 1 and "c""i" + 1 for any value of "i" and "c"1< "c"2.

Let "A""n" be the number of alternating permutations of the set {1, ..., "n"}. Then the exponential generating function of this sequence of numbers is a trigonometric function:

:sum_{n=0}^infty A_n {x^n over n!} = sec(x) + an(x) = anleft({x over 2} + {pi over 4} ight).

Consequently the numbers "A"2"n" with even indices are called secant numbers and those with odd indices are called tangent numbers.

ee also

* Boustrophedon transform

References

* André, D. "Développements de sec "x" et tan "x"." "Comptes Rendus Acad. Sci.", Paris 88, 965-967, 1879.
* André, D. "Mémoire sur les permutations alternées." "J. Math." 7, 167-184, 1881.


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