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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.

%I A200068 #31 Apr 25 2022 13:25:58
%S A200068 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,
%T A200068 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,
%U A200068 6,2,1,2,1,2,3,0,0,2,1,0,0,0,0,2
%N 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.
%H A200068 Alois P. Heinz, <a href="/A200068/b200068.txt">Rows n = 0..28, flattened</a>
%e A200068 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].
%e A200068 Irregular triangle begins:
%e A200068    1;
%e A200068    1;
%e A200068    2;
%e A200068    3, 1;
%e A200068    5, 1, 1, 1;
%e A200068    7, 3, 1, 3, 0, 0, 2;
%e A200068   11, 4, 2, 4, 3, 1, 3, 0, 1, 1, 1, 0, 1;
%e A200068   15, 8, 3, 8, 3, 3, 7, 1, 2, 3, 1, 3, 2, 0, 1, 2, 0, 0, 1, 0, 1;
%e A200068   ...
%p A200068 T:= proc(n) option remember; local mx, b, p;
%p A200068       b:=proc(m, i, l) local h;
%p A200068            if m=0 then p(i):= p(i)+1; if i>mx then mx:=i fi
%p A200068          else seq(b(m-h, i +add(`if`(l[j]<h, j*(h-l[j]), 0),
%p A200068                   j=1..nops(l)), [h, l[]]), h=1..m) fi
%p A200068          end;
%p A200068       mx:=0;
%p A200068       p:= proc() 0 end; forget(p);
%p A200068       b(n, 0, []); seq(p(i), i=0..mx)
%p A200068     end:
%p A200068 seq(T(n), n=0..10);
%t A200068 T[n_] := T[n] = Module[{mx, b, p},
%t A200068      b[m_, i_, l_] := Module[{h},
%t A200068           If[m == 0, p[i] = p[i]+1; If[i > mx, mx = i],
%t A200068           Table[b[m-h, i + Sum[If[l[[j]] < h, j*(h - l[[j]]), 0],
%t A200068                {j, 1, Length[l]}], Join[{h}, l]], {h, 1, m}]]];
%t A200068      mx = 0;
%t A200068      p[_] = 0;
%t A200068      b[n, 0, {}]; Table[p[i], {i, 0, mx}]];
%t A200068 Table[T[n], {n, 0, 10}] // Flatten (* _Jean-François Alcover_, Apr 25 2022, after _Alois P. Heinz_ *)
%Y A200068 Cf. A000041 (column k=0), A024786(n-1) (column k=1), A011782 (row sums), A200067 (row lengths -1), A189074.
%K A200068 nonn,tabf,look
%O A200068 0,3
%A A200068 _Alois P. Heinz_, Nov 13 2011