Alternating Permutation

An alternating permutation is an arrangement of the elements c_1, ..., c_n such that no element c_i has a magnitude between c_(i-1) and c_(i+1) is called an alternating (or zigzag) permutation. The determination of the number of alternating permutations for the set of the first n integers {1,2,...,n} is known as André's problem.

The numbers Z_n of alternating permutations on the integers from 1 to n for n=1, 2, ... are 1, 2, 4, 10, 32, 122, 544, ... (OEIS A001250). For example, the alternating permutations on n integers for small n are summarized in the following table.

nZ_nalternating permutations
22{1,2}, {2,1}
34{1,3,2}, {2,1,3}, {2,3,1}, {3,1,2}
410{1,3,2,4}, {1,4,2,3}, {2,1,4,3}, {2,3,1,4}, {2,4,1,3},
{3,1,4,2}, {3,2,4,1}, {3,4,1,2}, {4,1,3,2}, {4,2,3,1}

For n>1, every alternating permutation Z_n can be written either forward or reversed, and so must be an even number A_n=Z_n/2. The quantity A_n can be simply computed from the recurrence equation


where r and s pass through all integral numbers such that


a_0=a_1=1, and


The numbers A_n are sometimes called the Euler zigzag numbers, and the first few are given by 1, 1, 1, 2, 5, 16, 61, 272, ... (OEIS A000111).

The even-numbered A_ns are called Euler numbers |E_(2n)|, secant numbers S_n, or zig numbers (1, 1, 5, 61, 1385, ...; OEIS A000364), and the odd-numbered ones are sometimes called tangent numbers T_n or zag numbers (1, 2, 16, 272, 7936, ...; OEIS A000182).

Curiously, the secant and tangent Maclaurin series can be written in terms of the A_ns as


or combining them,


See also

Entringer Number, Euler Number, Euler Zigzag Number, Secant Number, Seidel-Entringer-Arnold Triangle, Tangent Number

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Referenced on Wolfram|Alpha

Alternating Permutation

Cite this as:

Weisstein, Eric W. "Alternating Permutation." From MathWorld--A Wolfram Web Resource.

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