OFFSET
0,3
COMMENTS
Row sums are powers of 2.
The Heubach et al. reference has a table for n <= 12.
LINKS
Alois P. Heinz, Rows n = 0..25, flattened
S. Heubach, A. Knopfmacher, M. E. Mays and A. Munagi, Inversions in Compositions of Integers, to appear in Quaestiones Mathematicae.
EXAMPLE
T(4,0) = 5: [4], [1,3], [2,2], [1,1,2], [1,1,1,1] - all partitions of 4.
T(5,2) = 3: [2,2,1], [3,1,1], [1,2,1,1].
T(6,4) = 2: [2,2,1,1], [2,1,1,1,1].
Triangle begins:
1
1
2
3 1
5 2 1
7 5 3 1
11 8 7 4 2
15 15 14 10 6 3 1
22 23 26 21 17 10 6 2 1
...
MAPLE
T:= proc(n) option remember; local b, p;
b:=proc(m, i, l)
if m=0 then p(i):= p(i)+1
else seq(b(m-h, i+nops(select(j->j<h, l)), [h, l[]]), h=1..m)
fi
end;
p:= proc() 0 end; forget(p);
b(n, 0, []); seq(p(i), i=0..floor(n^2/8))
end:
seq(T(n), n=0..12); # Alois P. Heinz, Apr 17 2011
MATHEMATICA
T[n_] := T[n] = Module[{b, p}, b[m_, i_, l_List] := If[m == 0, p[i] = p[i] + 1, Table[b[m-h, i+Length[Select[ l, #<h&]], Join[{h}, l]], {h, 1, m}]]; Clear[p]; p[_]=0; b[n, 0, {}]; Table[p[i], {i, 0, Floor[n^2/8]}]]; Table[ T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jan 17 2016, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
N. J. A. Sloane, Apr 16 2011
STATUS
approved