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.

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

Offset

0,3

Links

Alois P. Heinz, Rows n = 0..28, flattened

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

Adjacent sequences: A200065 A200066 A200067 * A200069 A200070 A200071

Keyword

nonn,tabf,look

Author

Alois P. Heinz, Nov 13 2011

Status

approved