OFFSET
0,9
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, Rows n = 0..140, flattened
FORMULA
Sum_{k=0..n} T(n,k) = T(n+1,n+1) = A291685(n).
T(2n,n) = T(2n,n+1) for all n>0.
EXAMPLE
T(3,1) = 1: 123.
T(3,2) = 2: 213, 231.
T(3,3) = 2: 312, 321.
Triangle T(n,k) begins:
1;
0, 1;
0, 1, 1;
0, 1, 2, 2;
0, 1, 5, 5, 5;
0, 1, 9, 12, 14, 16;
0, 1, 17, 36, 36, 47, 52;
0, 1, 31, 81, 98, 117, 166, 189;
0, 1, 57, 174, 327, 327, 425, 627, 683;
0, 1, 101, 413, 788, 988, 1116, 1633, 2400, 2621;
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:
T:= (n, k)-> b(0, n, k)-`if`(k=0, 0, b(0, n, k-1)):
seq(seq(T(n, k), k=0..n), n=0..12);
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]}]];
T[n_, k_] := b[0, n, k] - If[k == 0, 0, b[0, n, k - 1]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-Fran�ois Alcover, Jul 09 2018, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Aug 29 2017
STATUS
approved