A303287 Number of permutations p of [n] such that the sequence of ascents and descents of p or of p0 (if n is even) forms a Dyck path.

1, 1, 1, 2, 8, 22, 172, 604, 7296, 31238, 518324, 2620708, 55717312, 325024572, 8460090160, 55942352184, 1726791794432, 12765597850950, 456440969661508, 3730771315561300, 151770739970889792, 1359124435588313876, 62022635037246022000, 603916464771468176392

Offset

0,4

Links

Alois P. Heinz, Table of n, a(n) for n = 0..400

Wikipedia, Counting lattice paths

Formula

a(2n) = A303284(2n).

Example

a(0) = 1: the empty permutation.
a(1) = 1: 1.
a(2) = 1: 12.
a(3) = 2: 132, 231.
a(4) = 8: 1243, 1324, 1342, 1423, 2314, 2341, 2413, 3412.
a(5) = 22: 12543, 13254, 13542, 14253, 14352, 14532, 15243, 15342, 23154, 23541, 24153, 24351, 24531, 25143, 25341, 34152, 34251, 34521, 35142, 35241, 45132, 45231.

Maple

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

Mathematica

b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, 1, If[t > 0, Sum[b[u - j, o + j - 1, t - 1], {j, 1, u}], 0] + If[o + u > t, Sum[b[u + j - 1, o - j, t + 1], {j, 1, o}], 0]];
a[n_] := b[n, 0, 1];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, May 25 2018, translated from Maple *)

Crossrefs

Bisections give: A303285 (even part), A177042 (odd part).

Cf. A180056, A304001, A303284, A304777, A304778.

Sequence in context: A191643 A151356 A238530 * A321068 A158466 A065694

Adjacent sequences: A303284 A303285 A303286 * A303288 A303289 A303290

Keyword

nonn

Author

Alois P. Heinz, Apr 20 2018

Status

approved