OFFSET
0,9
LINKS
Alois P. Heinz, Rows n = 0..20, flattened
EXAMPLE
T(4,1) = 1: 1|2|3|4.
T(4,2) = 4: 12|34, 12|3|4, 1|23|4, 1|2|34.
T(4,3) = 5: 123|4, 13|24, 13|2|4, 1|234, 1|24|3.
T(4,4) = 5: 1234, 124|3, 134|2, 14|23, 14|2|3.
T(5,4) = 17: 1234|5, 124|35, 124|3|5, 134|25, 134|2|5, 13|245, 13|25|4, 14|235, 14|23|5, 1|2345, 1|235|4, 14|25|3, 14|2|35, 14|2|3|5, 1|245|3, 1|25|34, 1|25|3|4.
Triangle T(n,k) begins:
1;
0, 1;
0, 1, 1;
0, 1, 2, 2;
0, 1, 4, 5, 5;
0, 1, 7, 12, 17, 15;
0, 1, 12, 29, 45, 64, 52;
0, 1, 20, 66, 121, 201, 265, 203;
0, 1, 33, 145, 336, 585, 966, 1197, 877;
...
MAPLE
b:= proc(n, m, l) option remember; `if`(n=0, 1,
add(b(n-1, max(m, j), [subsop(1=NULL, l)[],
`if`(j<=m, 0, j)]), j={l[], m+1} minus {0}))
end:
A:= (n, k)-> `if`(n=0, 1, `if`(k<2, k, b(n, 0, [0$(k-1)]))):
T:= (n, k)-> A(n, k) -`if`(k=0, 0, A(n, k-1)):
seq(seq(T(n, k), k=0..n), n=0..12);
MATHEMATICA
b[n_, m_, l_List] := b[n, m, l] = If[n == 0, 1, Sum[b[n - 1, Max[m, j], Append[ReplacePart[l, 1 -> Nothing], If[j <= m, 0, j]]], {j, Append[l, m + 1] ~Complement~ {0}}]]; A[n_, k_] := If[n == 0, 1, If[k < 2, k, b[n, 0, Array[0&, k - 1]]]]; T [n_, k_] := A[n, k] - If[k == 0, 0, A[n, k - 1]]; Table[T[n, k], {n, 0, 12}, { k, 0, n}] // Flatten (* Jean-François Alcover, Feb 04 2017, translated from Maple *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Sep 16 2016
STATUS
approved