A287328 Number of permutations of [n] with exactly n (possibly overlapping) occurrences of the generalized pattern 13-2 (alternatively: 2-13, 2-31, or 31-2).

1, 0, 0, 0, 0, 0, 2, 42, 716, 11115, 166144, 2440438, 35573362, 517145226, 7517489564, 109427233994, 1596263948916, 23344423478564, 342325226476816, 5033781463313268, 74223067319020872, 1097343555359525542, 16265399526715750734, 241690082996256941332

Offset: 0

Author

Alois P. Heinz, Aug 31 2017

Keywords

• nonn

Comments

Also number of permutations of [n] with n nestings; also number of permutations of [n] with n crossings.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..300

Crossrefs

Main diagonal of A263776.

Programs

Maple

b:= proc(u, o) option remember;
      `if`(u+o=0, 1, add(b(u-j, o+j-1), j=1..u)+
       add(expand(b(u+j-1, o-j)*x^(j-1)), j=1..o))
    end:
a:= n-> coeff(b(n, 0), x, n):
seq(a(n), n=0..30);

Mathematica

b[u_, o_] := b[u, o] = If[u + o == 0, 1, Sum[b[u - j, o + j - 1], {j, 1, u}] + Sum[Expand[b[u + j - 1, o - j]*x^(j - 1)], {j, 1, o}]];
a[n_] := Coefficient[b[n, 0], x, n];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Dec 06 2023, after Alois P. Heinz *)

Formula

a(n) = A263776(n,n).