A303204 Number of permutations p of [n] such that 0p has a nonincreasing jump sequence beginning with ceiling(n/2).

1, 1, 1, 2, 5, 12, 36, 98, 327, 988, 3392, 10872, 38795, 129520, 469662, 1609176, 5935728, 20786804, 77416352, 274792342, 1035050705, 3719296036, 14094000938, 51119572738, 195075365778, 712918642042, 2734475097609, 10055531355652, 38747262233793

Offset: 0

Author

Alois P. Heinz, Apr 19 2018

Keywords

• nonn

Comments

An up-jump j occurs at position i in p if p_{i} > p_{i-1} and j is the index of p_i in the increasingly sorted list of those elements in {p_{i}, ..., p_{n}} that are larger than p_{i-1}. A down-jump j occurs at position i in p if p_{i} < p_{i-1} and j is the index of p_i in the decreasingly sorted list of those elements in {p_{i}, ..., p_{n}} that are smaller than p_{i-1}. First index in the lists is 1 here.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..500

Crossrefs

Bisections give A291688 (even part), A303203 (odd part).

Cf. A291684.

Programs

Maple

b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
      add(b(sort([u-j, o+j-1])[], j), j=1..min(t, u))+
      add(b(sort([u+j-1, o-j])[], j), j=1..min(t, o)))
    end:
a:= n-> `if`(n=0, 1, (j-> b(0, n, j)-b(0, n, j-1))(ceil(n/2))):
seq(a(n), n=0..30);

Mathematica

b[u_, o_, t_] := b[u, o, t] = If[u+o == 0, 1,
     Sum[b[Sequence @@ Sort[{u-j, o+j-1}], j], {j, Min[t, u]}]+
     Sum[b[Sequence @@ Sort[{u+j-1, o-j}], j], {j, Min[t, o]}]];
a[n_] := If[n == 0, 1,
     Function[j, b[0, n, j] - b[0, n, j-1]][Ceiling[n/2]]];
Table[a[n], {n, 0, 30}]; (* Jean-François Alcover, Aug 30 2021, after Alois P. Heinz *)

Formula

a(n) = A291684(n,ceiling(n/2)).