The number of Permutations of length with Runs, denoted
, , or . The Eulerian numbers are given explicitly by the sum

(1) |

(2) | |||

(3) |

together with the Recurrence Relation

(4) |

(5) |

The Eulerian numbers satisfy

(6) |

(7) |

**References**

Carlitz, L. ``Eulerian Numbers and Polynomials.'' *Math. Mag.* **32**, 247-260, 1959.

Foata, D. and Schützenberger, M.-P. *Théorie Géométrique des Polynômes Eulériens.* Berlin: Springer-Verlag, 1970.

Kimber, A. C. ``Eulerian Numbers.'' Supplement to *Encyclopedia of Statistical Sciences.* (Eds. S. Kotz,
N. L. Johnson, and C. B. Read). New York: Wiley, pp. 59-60, 1989.

Salama, I. A. and Kupper, L. L. ``A Geometric Interpretation for the Eulerian Numbers.'' *Amer. Math. Monthly*
**93**, 51-52, 1986.

Sloane, N. J. A. Sequence A008292 in ``The On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.

© 1996-9

1999-05-25