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A238343
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Triangle T(n,k) read by rows: T(n,k) is the number of compositions of n with k descents, n>=0, 0<=k<=n.
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28
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1, 1, 0, 2, 0, 0, 3, 1, 0, 0, 5, 3, 0, 0, 0, 7, 9, 0, 0, 0, 0, 11, 19, 2, 0, 0, 0, 0, 15, 41, 8, 0, 0, 0, 0, 0, 22, 77, 29, 0, 0, 0, 0, 0, 0, 30, 142, 81, 3, 0, 0, 0, 0, 0, 0, 42, 247, 205, 18, 0, 0, 0, 0, 0, 0, 0, 56, 421, 469, 78, 0, 0, 0, 0, 0, 0, 0, 0, 77, 689, 1013, 264, 5, 0, 0, 0, 0, 0, 0, 0, 0, 101, 1113, 2059, 786, 37, 0, 0, 0, 0, 0, 0, 0, 0, 0
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OFFSET
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0,4
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COMMENTS
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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
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LINKS
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EXAMPLE
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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;
...
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)
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MAPLE
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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);
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MATHEMATICA
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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 *)
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CROSSREFS
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Columns k=0-10 give: A000041, A241626, A241627, A241628, A241629, A241630, A241631, A241632, A241633, A241634, A241635.
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 without zeros is A238344.
The version for weak ascents is A333213.
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KEYWORD
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AUTHOR
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STATUS
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approved
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