A231228 Number of permutations of [n] with exactly one occurrence of one of the consecutive patterns 123, 1432, 2431, 3421.

0, 0, 0, 1, 9, 59, 358, 2235, 14658, 103270, 778451, 6315499, 54733657, 507655301, 5003179539, 52430810493, 580611272956, 6796733911852, 83658527086447, 1083027034959367, 14678725047527255, 208344799726820123, 3084495765476262875, 47646333262275943521

Offset: 0

Author

Alois P. Heinz, Nov 05 2013

Keywords

• nonn

Examples

			a(3) = 1: 123.
a(4) = 9: 1243, 1342, 1432, 2134, 2341, 2431, 3124, 3421, 4123.
a(5) = 59: 12435, 12534, 13245, ..., 53124, 53421, 54123.
a(6) = 358: 124365, 125364, 125463, ..., 653124, 653421, 654123.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200

Crossrefs

Column k=1 of A231210.

Programs

Maple

b:= proc(u, o, t) option remember;
      `if`(t=7, 0, `if`(u+o=0, `if`(t in [4, 5, 6], 1, 0),
      add(b(u+j-1, o-j, [2, 5, 2, 5, 7, 5][t]), j=1..o)+
      add(b(u-j, o+j-1, [1, 3, 4, 4, 6, 7][t]), j=1..u)))
    end:
a:= n-> b(n, 0, 1):
seq(a(n), n=0..25);

Mathematica

b[u_, o_, t_] := b[u, o, t] = If[t==7, 0, If[u+o==0, If[4 <= t <= 6, 1, 0],
      Sum[b[u + j - 1, o - j, {2, 5, 2, 5, 7, 5}[[t]]], {j, 1, o}] +
      Sum[b[u - j, o + j - 1, {1, 3, 4, 4, 6, 7}[[t]]], {j, 1, u}]]];
a[n_] := b[n, 0, 1];
a /@ Range[0, 25] (* Jean-François Alcover, Jan 03 2021, after Alois P. Heinz *)

Formula

a(n) ~ c * (2/Pi)^n * n! * n, where c = 3.08472832460941829086964816782... . - Vaclav Kotesovec, Aug 28 2014