Eulerian Number

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

 (1)

Making the definition
 (2) (3)

together with the Recurrence Relation
 (4)

for then gives
 (5)

The arrangement of the numbers into a triangle gives Euler's Triangle, whose entries are 1, 1, 1, 1, 4, 1, 1, 11, 11, 1, ... (Sloane's A008292). Therefore, they represent a sort of generalization of the Binomial Coefficients where the defining Recurrence Relation weights the sum of neighbors by their row and column numbers, respectively.

The Eulerian numbers satisfy

 (6)

Eulerian numbers also arise in the surprising context of integrating the Sinc Function, and also in sums of the form
 (7)

where is the Polylogarithm function.

See also Combination Lock, Euler Number, Euler's Triangle, Euler Zigzag Number, Polylogarithm, Sinc Function, Worpitzky's Identity, Z-Transform

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.