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Entringer Number

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1 0 1 0 1 1 0 1 2 2 0 2 4 5 5
(1)

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)=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

A_n={i^nE_n for n even; -((2i)^(n+1)(2^(n+1)-1)B_(n+1))/(n+1) for n odd,
(6)

. 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 Number

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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."

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Entringer Number

Cite this as:

Weisstein, Eric W. "Entringer Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EntringerNumber.html

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