A303203 Number of permutations p of [2n+1] such that 0p has a nonincreasing jump sequence beginning with n+1.

1, 2, 12, 98, 988, 10872, 129520, 1609176, 20786804, 274792342, 3719296036, 51119572738, 712918642042, 10055531355652, 143287150725298, 2058368140071146, 29796314629826814, 434051906728752164, 6359063811707227298, 93635249115751389952, 1385028309353547034876

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..250

Crossrefs

Bisection (odd part) of A303204.

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-> b(0, 2*n+1, n+1)-b(0, 2*n+1, n):
seq(a(n), n=0..25);

Mathematica

b[u_, o_, t_] := b[u, o, t] = If[u+o == 0, 1,
     Sum[b[u-j, o+j-1, j], {j, 1, Min[t, u]}] +
     Sum[b[u+j-1, o-j, j], {j, 1, Min[t, o]}]];
a[n_] := b[0, 2n+1, n+1] - b[0, 2n+1, n];
a /@ Range[0, 25] (* Jean-François Alcover, Sep 01 2021, after Alois P. Heinz *)

Formula

a(n) = A291684(2n+1,n+1).