Alternating permutation

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In combinatorial mathematics, an alternating permutation of the set is an arrangement of those numbers into an order c₁, ..., c_n such that no element c_i is between c_i − 1 and c_i + 1 for any value of i.

Let A_n be the number of alternating permutations of the set . Then the exponential generating function of this sequence of numbers is a trigonometric function:

[\sum_^\infty A_n = \sec(x) + \tan(x) = \tan\left( + \right).]

Consequently the numbers A_2n with even indices are called secant numbers and those with odd indices are called tangent numbers.

See also

• Boustrophedon transform

References

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

 

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