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# Alternating Permutation

An alternating permutation is an arrangement of the elements , ..., such that no element has a magnitude between and is called an alternating (or zigzag) permutation. The determination of the number of alternating permutations for the set of the first integers is known as André's problem.

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

 alternating permutations 1 1 2 2 , 3 4 , , , 4 10 , , , , , , , , ,

For , every alternating permutation can be written either forward or reversed, and so must be an even number . The quantity can be simply computed from the recurrence equation

 (1)

where and pass through all integral numbers such that

 (2)

, and

 (3)

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

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

 (4) (5)

or combining them,

 (6)

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

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## References

André, D. "Developments de et ." Comptes Rendus Acad. Sci. Paris 88, 965-967, 1879.André, D. "Memoire sur les permutations alternées." J. Math. 7, 167-184, 1881.Arnold, V. I. "Bernoulli-Euler Updown Numbers Associated with Function Singularities, Their Combinatorics and Arithmetics." Duke Math. J. 63, 537-555, 1991.Arnold, V. I. "Snake Calculus and Combinatorics of Bernoulli, Euler, and Springer Numbers for Coxeter Groups." Russian Math. Surveys 47, 3-45, 1992.Bauslaugh, B. and Ruskey, F. "Generating Alternating Permutations Lexicographically." BIT 30, 17-26, 1990.Conway, J. H. and Guy, R. K. In The Book of Numbers. New York: Springer-Verlag, pp. 110-111, 1996.Dörrie, H. "André's Derivation of the Secant and Tangent Series." §16 in 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, pp. 64-69, 1965.Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., pp. 69-75, 1985.Knuth, D. E. and Buckholtz, T. J. "Computation of Tangent, Euler, and Bernoulli Numbers." Math. Comput. 21, 663-688, 1967.Millar, J.; Sloane, N. J. A.; and Young, N. E. "A New Operation on Sequences: The Boustrophedon Transform." J. Combin. Th. Ser. A 76, 44-54, 1996.Ruskey, F. "Information of Alternating Permutations." http://www.theory.csc.uvic.ca/~cos/inf/perm/Alternating.html.Sloane, N. J. A. Sequences A000111/M1492, A000182/M2096, A000364/M4019, and A001250/M1235 in "The On-Line Encyclopedia of Integer Sequences."

## Referenced on Wolfram|Alpha

Alternating Permutation

## Cite this as:

Weisstein, Eric W. "Alternating Permutation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/AlternatingPermutation.html