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..100
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Sep 01 2017
STATUS
approved