Entringer Number
The Entringer numbers E(n,k) (OEIS A008281) are the number of permutations of {1,2,...,n+1}, starting with k+1, which, after initially falling, alternately fall then rise. The Entringer numbers are given by
E(0,0) = 1
(2)
E(n,0) =
(3)
together with the recurrence relation
| E(n,k)=E(n,k-1)+E(n-1,n-k). |
(4)
|
A suitably arranged number triangle of these numbers is known as the Seidel-Entringer-Arnold triangle.
The numbers A(n)=E(n,n) are the secant and tangent numbers given by the Maclaurin series
| secx+tanx=A_0+A_1x+A_2(x^2)/(2!)+A_3(x^3)/(3!)+A_4(x^4)/(4!)+A_5(x^5)/(5!)+.... |
(5)
|
They have closed form
. where E_n is an Euler number and B_n is a Bernoulli number.
See also
Alternating Permutation, Boustrophedon Transform, Euler Zigzag Number, Permutation, Secant Number, Seidel-Entringer-Arnold Triangle, Tangent Number, Zag Number, Zig NumberExplore with Wolfram|Alpha
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References
Bauslaugh, B. and Ruskey, F. "Generating Alternating Permutations Lexographically." BIT 80, 17-26, 1990.Entringer, R. C. "A Combinatorial Interpretation of the Euler and Bernoulli Numbers." Nieuw Arch. Wisk. 14, 241-246, 1966.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.Poupard, C. "De nouvelles significations enumeratives des nombres d'Entringer." Disc. Math. 38, 265-271, 1982.Ruskey, F. "Information of Alternating Permutations." http://www.theory.csc.uvic.ca/~cos/inf/perm/Alternating.html.Sloane, N. J. A. Sequences A000111/M1492 and A008281 in "The On-Line Encyclopedia of Integer Sequences."Referenced on Wolfram|Alpha
Entringer NumberCite this as:
Weisstein, Eric W. "Entringer Number." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/EntringerNumber.html