A220589 Number of simple skew-merged permutations with n elements.

2, 2, 8, 16, 44, 108, 284, 740, 1966, 5254, 14172, 38476, 105122, 288754, 797036, 2209588, 6149618, 17176186, 48129284, 135261796, 381169532, 1076824852, 3049109912, 8652239496, 24600592454, 70075316198, 199955694616

Offset: 4

Author

Vincent Vatter, Jan 14 2013

Keywords

• nonn
• easy

Comments

A permutation is skew-merged if it is the union of an increasing subsequence and a decreasing subsequence. A permutation is simple if it does not contain a nontrivial interval.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 4..1000
• Michael H. Albert and Vincent Vatter, Generating and enumerating 321-avoiding and skew-merged simple permutations

Crossrefs

Cf. A029759.

Programs

Mathematica

Rest[Rest[Rest[Rest[CoefficientList[Series[(1-2*x-x^2+(x-1)*Sqrt[1-2*x-3*x^2])/(x+1), {x, 0, 20}], x]]]]] (* Vaclav Kotesovec, Jan 16 2013 *)

Formula

G.f.: (1-2*x-x^2+(x-1)*sqrt(1-2*x-3*x^2)) / (x+1).

Recurrence (for n>4): (n-4)*n*a(n) = (n^2-7*n+15)*a(n-1) + (n-3)*(5*n-17)*a(n-2) + 3*(n-4)*(n-3)*a(n-3). - Vaclav Kotesovec, Jan 16 2013

a(n) ~ 3^(n-1/2)/(2*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jan 16 2013