Alternating permutation

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In combinatorial mathematics, an alternating permutation of the set {1, 2, 3, ..., n} is an arrangement of those numbers into an order c1, ..., cn such that no element ci is between ci − 1 and ci + 1 for any value of i and c1< c2.

Let An 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) + \tan(x) = \tan\left({x \over 2} + {\pi \over 4}\right).

Consequently the numbers A2n with even indices are called secant numbers and those with odd indices are called tangent numbers.

[edit] See also

[edit] 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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