OFFSET
0,4
COMMENTS
Counting ascents gives the same triangle.
For n > 0, also the number of compositions of n with k + 1 maximal weakly increasing runs. - Gus Wiseman, Mar 23 2020
LINKS
Joerg Arndt and Alois P. Heinz, Rows n = 0..140, flattened
EXAMPLE
Triangle starts:
00: 1;
01: 1, 0;
02: 2, 0, 0;
03: 3, 1, 0, 0;
04: 5, 3, 0, 0, 0;
05: 7, 9, 0, 0, 0, 0;
06: 11, 19, 2, 0, 0, 0, 0;
07: 15, 41, 8, 0, 0, 0, 0, 0;
08: 22, 77, 29, 0, 0, 0, 0, 0, 0;
09: 30, 142, 81, 3, 0, 0, 0, 0, 0, 0;
10: 42, 247, 205, 18, 0, 0, 0, 0, 0, 0, 0;
11: 56, 421, 469, 78, 0, 0, 0, 0, 0, 0, 0, 0;
12: 77, 689, 1013, 264, 5, 0, 0, 0, 0, 0, 0, 0, 0;
13: 101, 1113, 2059, 786, 37, 0, 0, 0, 0, 0, 0, 0, 0, 0;
14: 135, 1750, 4021, 2097, 189, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;
15: 176, 2712, 7558, 5179, 751, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;
...
From Gus Wiseman, Mar 23 2020: (Start)
Row n = 5 counts the following compositions:
(5) (3,2)
(1,4) (4,1)
(2,3) (1,3,1)
(1,1,3) (2,1,2)
(1,2,2) (2,2,1)
(1,1,1,2) (3,1,1)
(1,1,1,1,1) (1,1,2,1)
(1,2,1,1)
(2,1,1,1)
(End)
MAPLE
b:= proc(n, i) option remember; `if`(n=0, 1, expand(
add(b(n-j, j)*`if`(j<i, x, 1), j=1..n)))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 0)):
seq(T(n), n=0..20);
MATHEMATICA
b[n_, i_] := b[n, i] = If[n == 0, 1, Sum[b[n-j, j]*If[j<i, x, 1], {j, 1, n}]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, 0]]; Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, Jan 08 2015, translated from Maple *)
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], n==0||Length[Split[#, LessEqual]]==k+1&]], {n, 0, 9}, {k, 0, n}] (* Gus Wiseman, Mar 23 2020 *)
CROSSREFS
Columns k=0-10 give: A000041, A241626, A241627, A241628, A241629, A241630, A241631, A241632, A241633, A241634, A241635.
T(3n,n) gives A000045(n+1).
T(3n+1,n) = A136376(n+1).
Row sums are A011782.
Compositions by length are A007318.
The version for co-runs or levels is A106356.
The case of partitions (instead of compositions) is A133121.
The version for runs is A238279.
The version without zeros is A238344.
The version for weak ascents is A333213.
KEYWORD
nonn,tabl
AUTHOR
Joerg Arndt and Alois P. Heinz, Feb 25 2014
STATUS
approved