A263885 Number of permutations of [n] containing exactly one occurrence of the consecutive pattern 132.

1, 8, 54, 368, 2649, 20544, 172596, 1569408, 15398829, 162412416, 1834081890, 22093090560, 282889238253, 3837991053312, 55010010678120, 830731742908416, 13185328329110745, 219457733809563648, 3822426663111579150, 69538569862816419840, 1318999546575572747265

Offset: 3

Author

Alois P. Heinz, Oct 28 2015

Keywords

• nonn

Examples

			a(3) = 1: 132.
a(4) = 8: 1243, 1324, 1423, 1432, 2143, 2431, 3142, 4132.
a(5) = 54: 12354, 12435, 12534, ..., 52431, 53142, 54132.
a(6) = 368: 123465, 123546, 123645, ..., 652431, 653142, 654132.
a(7) = 2649: 1234576, 1234657, 1234756, ..., 7652431, 7653142, 7654132.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 3..200
• Eric Weisstein's World of Mathematics, Inverse Erf

Crossrefs

Column k=1 of A197365.

Programs

Maple

b:= proc(u, o, t, c) option remember; `if`(u+o=0, c, add(
      b(u-j, o+j-1, 0, c+`if`(j<=t, 1, 0)), j=`if`(c=1, t, 0)
      +1..u) +add(b(u+j-1, o-j, j-1, c), j=1..o))
    end:
a:= n-> b(n, 0$3):
seq(a(n), n=3..30);

Mathematica

Drop[Coefficient[CoefficientList[Series[1/(1 - (Sqrt[Pi/2]*Erfi[(Sqrt[u-1]*x) / Sqrt[2]])/Sqrt[u-1]), {x, 0, 25}], x] * Range[0, 25]!, u], 3] (* Vaclav Kotesovec, Oct 29 2015 *)

Formula

a(n) = A197365(n,1).

a(n) ~ c * d^n * n! * n, where d = 1/A240885 = 1/(sqrt(2) * InverseErf(sqrt(2/Pi))) = 0.78397693120354749... and c = 0.679554202696108785... . - Vaclav Kotesovec, Oct 29 2015