A288910 Number of permutations p of [n] such that 0p has a nonincreasing up-jump sequence and also has a nonincreasing down-jump sequence.

1, 1, 2, 5, 18, 69, 303, 1357, 6552, 31961, 163587, 839710, 4485686, 23917300, 131366017, 722130351, 4059017880, 22809880116, 130713878478, 748596353814, 4354695496124, 25349532110660, 149415724294027, 881419904003486, 5256588077063477, 31377362011756061

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..100

Crossrefs

Cf. A288911, A288912, A291685.

Programs

Maple

b:= proc(u, o, t, s) option remember; `if`(u+o=0, 1,
      add(b(u-j, o+j-1, j, s), j=1..min(t, u))+
      add(b(u+j-1, o-j, t, j), j=1..min(s, o)))
    end:
a:= n-> b(0, n$3):
seq(a(n), n=0..26);

Mathematica

b[u_, o_, t_, s_] := b[u, o, t, s] = If[u + o == 0, 1,
     Sum[b[u - j, o + j - 1, j, s], {j, Min[t, u]}] +
     Sum[b[u + j - 1, o - j, t, j], {j, Min[s, o]}]];
a[n_] := b[0, n, n, n];
Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Aug 30 2021, after Alois P. Heinz *)