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
Examples
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; ...
Links
- Alois P. Heinz, Rows n = 0..28, flattened
Crossrefs
Programs
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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] -
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 *)