OFFSET
0,3
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
FORMULA
a(n) = A291684(2n,n).
EXAMPLE
a(2) = 5: 2134, 2314, 2341, 2413, 2431.
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Aug 29 2017
STATUS
approved