A228422 Number of permutations of [n] with exactly two (possibly overlapping) occurrences of some of the consecutive patterns 123, 1432, 2431, 3421.

0, 0, 0, 0, 1, 14, 164, 1589, 15034, 139465, 1334945, 13108425, 134906641, 1443572465, 16238742806, 190546010823, 2347715040542, 30162115442344, 405859441345002, 5684963539755583, 83163913991455832, 1263763900212930657, 20000260465018111763

Offset: 0

Author

Alois P. Heinz, Nov 09 2013

Keywords

• nonn

Examples

			a(4) = 1: 1234.
a(5) = 14: 12354, 12453, 12543, 13452, 13542, 14532, 21345, 23451, 23541, 24531, 31245, 34521, 41235, 51234.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..185

Crossrefs

Column k=2 of A231210.

Programs

Maple

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

Mathematica

b[u_, o_, t_, c_] := b[u, o, t, c] =
    If[c<0, 0, If[u+o == 0, If[c == 0, 1, 0],
    Sum[b[u+j-1, o-j, 2,              If[t == 2, c-1, c]], {j, 1, o}] +
    Sum[b[u-j, o+j-1, {1, 3, 1}[[t]], If[t == 3, c-1, c]], {j, 1, u}]]];
a[n_] := b[n, 0, 1, 2];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Dec 20 2020, after Alois P. Heinz *)

Formula

a(n) ~ c * (2/Pi)^n * n! * n^2, where c = 1.286210080518397686... . - Vaclav Kotesovec, Aug 28 2014