A291688 Number of permutations p of [2n] such that 0p has a nonincreasing jump sequence beginning with n.

1, 1, 5, 36, 327, 3392, 38795, 469662, 5935728, 77416352, 1035050705, 14094000938, 195075365778, 2734475097609, 38747262233793, 554199475506095, 7990492729051526, 115995691148658656, 1694340616136589743, 24882428969673439384, 367160435328847044586

Offset: 0

Author

Alois P. Heinz, Aug 29 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.

Examples

			a(2) = 5: 2134, 2314, 2341, 2413, 2431.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..250

Crossrefs

Cf. A291684.

Bisection (even part) of A303204.

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, j), j=1..min(t, o)))
    end:
a:= n-> b(0, 2*n, n)-`if`(n=0, 0, b(0, 2*n, n-1)):
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, Min[t, u]}] +
     Sum[b[u + j - 1, o - j, j], {j, Min[t, o]}]];
a[n_] :=  b[0, 2n, n] - If[n == 0, 0, b[0, 2n, n - 1]];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Aug 30 2021, after Alois P. Heinz *)

Formula

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