A186735 Number of permutations of [n] with no ascending runs of length 1 or 2.

1, 0, 0, 1, 1, 1, 20, 69, 180, 1930, 12611, 61051, 566129, 5179750, 38348469, 376547340, 4169246332, 41559058969, 465750294781, 5905176350849, 72848728572828, 946103621115633, 13501160406995728, 195518567272213262, 2918439778172724571, 46559546190633191495

Offset: 0

Author

Alois P. Heinz, Aug 29 2013

Keywords

• nonn

Examples

			a(0) = 1: the empty permutation.
a(3) = 1: 123.
a(4) = 1: 1234.
a(5) = 1: 12345.
a(6) = 20: 123456, 124356, 125346, 126345, 134256, 135246, 136245, 145236, 146235, 156234, 234156, 235146, 236145, 245136, 246135, 256134, 345126, 346125, 356124, 456123.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..150

Crossrefs

Cf. A064315, A097899.

Cf. A000142, A185652, A228614.

Programs

Mathematica

A[n_, k_] := A[n, k] = Module[{b}, b[u_, o_, t_] := b[u, o, t] =
     If[t + o <= k, (u + o)!,
     Sum[b[u + i - 1, o - i, Min[k, t] + 1], {i, 1, o}] +
     If[t <= k, u*(u + o - 1)!,
     Sum[b[u - i, o + i - 1, 1], {i, 1, u}]]];
  Sum[b[j - 1, n - j, 1], {j, 1, n}]];
a[n_] := n! - A[n, 2];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Sep 03 2021, after Alois P. Heinz in A064315 *)

Formula

a(n) = A000142(n) - A228614(n) - A185652(n).