A288911 Number of permutations p of [n] such that 0p has a nonincreasing up-jump sequence.

1, 1, 2, 5, 19, 80, 416, 2306, 14588, 98053, 724183, 5633793, 47416901, 417050215, 3914152702, 38288228393, 395496623939, 4241350801439, 47715403637219, 555476398869869, 6744406721447538, 84548532634924758, 1100301545470162305, 14751287346427752887

Offset: 0

Author

Alois P. Heinz, Sep 01 2017

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..200

Crossrefs

Cf. A288910, A288912, A291685.

Programs

Maple

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

Mathematica

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