Entringer Number

The Entringer numbers $E(n,k)$ are the number of Permutations of $\{1, 2, \ldots, n+1\}$, starting with $k+1$, which, after initially falling, alternately fall then rise. The Entringer numbers are given by

$$E(0,0) = 1$$

$$E(n,0) = 0$$

together with the Recurrence Relation

$$E(n,k)=E(n,k+1)+E(n-1,n-k).$$

The numbers $E(n)=E(n,n)$ are the Secant and Tangent Numbers given by the Maclaurin Series

$$\sec x+\tan x = A_0+A_1 x+A_2 {x^2\over 2!}+A_3 {x^3\over 3!}+A_4 {x^4\over 4!}+A_5 {x^5\over 5!}+\ldots.$$

See also Alternating Permutation, Boustrophedon Transform, Euler Zigzag Number, Permutation, Secant Number, Seidel-Entringer-Arnold Triangle, Tangent Number, Zag Number, Zig Number

References

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.