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A200068 Irregular triangle read by rows: T(n,k), n>=0, 0<=k<=A200067(n), is number of compositions of n such that the sum of weighted inversions equals k and weights are products of absolute differences and distances between the element pairs which are not in sorted order. 2
1, 1, 2, 3, 1, 5, 1, 1, 1, 7, 3, 1, 3, 0, 0, 2, 11, 4, 2, 4, 3, 1, 3, 0, 1, 1, 1, 0, 1, 15, 8, 3, 8, 3, 3, 7, 1, 2, 3, 1, 3, 2, 0, 1, 2, 0, 0, 1, 0, 1, 22, 11, 7, 12, 4, 5, 13, 5, 4, 7, 4, 4, 5, 0, 3, 6, 2, 1, 2, 1, 2, 3, 0, 0, 2, 1, 0, 0, 0, 0, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
EXAMPLE
The compositions of n = 4 have weighted inversions 0: [4], [2,2], [1,3], [1,1,2], [1,1,1,1]; 1: [1,2,1]; 2: [3,1]; 3: [2,1,1]; => row 4 = [5,1,1,1].
Irregular triangle begins:
1;
1;
2;
3, 1;
5, 1, 1, 1;
7, 3, 1, 3, 0, 0, 2;
11, 4, 2, 4, 3, 1, 3, 0, 1, 1, 1, 0, 1;
15, 8, 3, 8, 3, 3, 7, 1, 2, 3, 1, 3, 2, 0, 1, 2, 0, 0, 1, 0, 1;
...
MAPLE
T:= proc(n) option remember; local mx, b, p;
b:=proc(m, i, l) local h;
if m=0 then p(i):= p(i)+1; if i>mx then mx:=i fi
else seq(b(m-h, i +add(`if`(l[j]<h, j*(h-l[j]), 0),
j=1..nops(l)), [h, l[]]), h=1..m) fi
end;
mx:=0;
p:= proc() 0 end; forget(p);
b(n, 0, []); seq(p(i), i=0..mx)
end:
seq(T(n), n=0..10);
MATHEMATICA
T[n_] := T[n] = Module[{mx, b, p},
b[m_, i_, l_] := Module[{h},
If[m == 0, p[i] = p[i]+1; If[i > mx, mx = i],
Table[b[m-h, i + Sum[If[l[[j]] < h, j*(h - l[[j]]), 0],
{j, 1, Length[l]}], Join[{h}, l]], {h, 1, m}]]];
mx = 0;
p[_] = 0;
b[n, 0, {}]; Table[p[i], {i, 0, mx}]];
Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Apr 25 2022, after Alois P. Heinz *)
CROSSREFS
Cf. A000041 (column k=0), A024786(n-1) (column k=1), A011782 (row sums), A200067 (row lengths -1), A189074.
Sequence in context: A187680 A328731 A140706 * A139764 A371356 A227643
KEYWORD
nonn,tabf,look
AUTHOR
Alois P. Heinz, Nov 13 2011
STATUS
approved

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Last modified March 28 08:02 EDT 2024. Contains 371236 sequences. (Running on oeis4.)