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) |

(2) |

(3) |

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

(4) | |||

(5) |

or combining them,

(6) |

**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.

© 1996-9

1999-05-25