A121703 - OEIS

A121703

Number of alternating separable permutations.

1, 2, 4, 8, 20, 48, 132, 344, 996, 2720, 8132, 22888, 69940, 201040, 624132, 1822136, 5725124, 16915008, 53648772, 160012232, 511360340, 1536928624, 4942300804, 14949122328, 48322714020, 146946942688, 477105960772, 1457491035944, 4750171491956, 14568377075344

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OFFSET

1,2

COMMENTS

The separable permutations are those avoiding 2413 and 3142 and are counted by the large Schroeder numbers (A006318). The alternating permutations are counted by the Euler numbers (A000111).

LINKS

Table of n, a(n) for n=1..30.

Robert Brignall, Sophie Huczynska, and Vincent Vatter, Simple permutations and algebraic generating functions, arXiv:math/0608391 [math.CO], 2006.

FORMULA

G.f. satisfies f^3-(2x^2-5x+4)f^2-(4x^3+x^2-8x)f-(2x^4+5x^3+4x^2)=0.

EXAMPLE

a(4)=8 because of the 10 alternating permutations of length 4, 2413 and 3142 are not separable.

MATHEMATICA

nmax = 40; aa = ConstantArray[0, nmax]; aa[[1]] = 1; aa[[2]] = 2; Do[f = Sum[aa[[k]]*x^k, {k, 1, j - 1}] + koef*x^j; sol = Solve[SeriesCoefficient[f^3 - (2*x^2 - 5*x + 4)*f^2 - (4*x^3 + x^2 - 8*x)*f - (2*x^4 + 5*x^3 + 4*x^2), {x, 0, j + 2}] == 0, koef][[1]]; aa[[j]] = koef /. sol[[1]], {j, 3, nmax}]; aa (* Vaclav Kotesovec, Jul 07 2024 *)

CROSSREFS

Cf. A121704.

Sequence in context: A056952 A369496 A225585 * A301467 A358056 A275070

Adjacent sequences: A121700 A121701 A121702 * A121704 A121705 A121706

KEYWORD

nonn

AUTHOR

Vincent Vatter, Aug 16 2006

EXTENSIONS

a(13) and beyond corrected by Vaclav Kotesovec, Jul 07 2024

STATUS

approved