## Alternating Permutation

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. An example of an alternating permutation is (1, 3, 2, 5, 4).

As many alternating permutations among elements begin by rising as by falling. The magnitude of the s does not matter; only the number of them. Let the number of alternating permutations be given by . This quantity can then be computed from

 (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, ... (Sloane's A000111). The Odd-numbered s are called Euler Numbers, Secant Numbers, or Zig Numbers, and the Even-numbered ones are sometimes called Tangent Numbers or Zag Numbers.

Curiously enough, the Secant and Tangent Maclaurin Series can be written in terms of the s as

 (4) (5)

or combining them,

 (6)

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

References

André, D. Developments de et .'' C. R. 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 Deviation 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://sue.csc.uvic.ca/~cos/inf/perm/Alternating.html.

Sloane, N. J. A. Sequence A000111/M1492 in An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.