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