A194547 Triangle read by rows: T(n,k) = Dyson's rank of the k-th partition of n, with partitions in lexicographic order.

0, -1, 1, -2, 0, 2, -3, -1, 1, 0, 3, -4, -2, 0, -1, 2, 1, 4, -5, -3, -1, -2, 1, 0, 3, -1, 2, 1, 5, -6, -4, -2, -3, 0, -1, 2, -2, 1, 0, 4, 0, 3, 2, 6, -7, -5, -3, -4, -1, -2, 1, -3, 0, -1, 3, -1, 2, 1, 5, -2, 1, 0, 4, 3, 2, 7, -8, -6, -4, -5, -2, -3, 0, -4, -1

Offset: 1

Author

Omar E. Pol, Dec 10 2011

Keywords

• sign
• tabf
• look

Comments

Row n has length A000041(n). The sum of row n is equal to zero.

Examples

			Written as a triangle:
  0;
  -1,1;
  -2,0,2;
  -3,-1,1,0,3;
  -4,-2,0,-1,2,1,4;
  -5,-3,-1,-2,1,0,3,-1,2,1,5;
  -6,-4,-2,-3,0,-1,2,-2,1,0,4,0,3,2,6;
  -7,-5,-3,-4,-1,-2,1,-3,0,-1,3,-1,2,1,5,-2,1,0,4,3,2,7;
		

Links

• Alois P. Heinz, Rows n = 1..26, flattened

Crossrefs

Cf. A105805, A135010, A138121, A194546, A194548, A194549.

Programs

Maple

T:= proc(n) local b, l;
      b:= proc(n, i, t)
            if n=0 then l:=l, i-t
          elif i>n then
          else b(n-i, i, t+1); b(n, i+1, t)
            fi
          end;
      l:= NULL; b(n, 1, 0); l
    end:
seq(T(n), n=1..10);  # Alois P. Heinz, Dec 22 2011

Mathematica

T[n_] := Module[{b, l}, b[n0_, i_, t_] := If [n0==0, l = Append[l, i-t], If[i>n0, , b[n0-i, i, t+1]; b[n0, i+1, t]]]; l = {}; b[n, 1, 0]; l];
Table[T[n], {n, 1, 10}] // Flatten (* Jean-François Alcover, Mar 05 2021, after Alois P. Heinz *)

Formula

a(n) = A194546(n) - A193173(n).

Extensions

More terms from Alois P. Heinz, Dec 22 2011