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

0,5

Links

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

Formula

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

Example

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.

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 *)

Crossrefs

Column k=1 of A231210.

Sequence in context: A174654 A027249 A026717 * A198847 A059356 A039929

Adjacent sequences: A231225 A231226 A231227 * A231229 A231230 A231231

Keyword

nonn

Author

Alois P. Heinz, Nov 05 2013

Status

approved