A288912 Number of permutations p of [n] such that 0p has a nonincreasing down-jump sequence.

1, 1, 2, 6, 23, 106, 558, 3284, 21200, 148539, 1119273, 9013112, 77106652, 697811164, 6652604804, 66593158893, 697756930786, 7633155361594, 86969814549075, 1029939820075074, 12652809025029242, 160977119684852369, 2117642963178349336, 28763717105362639324

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, A288911, A291685.

Programs

Maple

b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
      add(b(u-j, o+j-1, j), j=1..min(t, u))+
      add(b(u+j-1, o-j, t), j=1..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, j], {j, Min[t, u]}] +
     Sum[b[u + j - 1, o - j, t], {j, o}]];
a[n_] := b[0, n, n];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Aug 31 2021, after Alois P. Heinz *)