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

%I A194547 #31 Mar 05 2021 07:47:41
%S A194547 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,
%T A194547 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,
%U A194547 -1,2,1,5,-2,1,0,4,3,2,7,-8,-6,-4,-5,-2,-3,0,-4,-1
%N A194547 Triangle read by rows: T(n,k) = Dyson's rank of the k-th partition of n, with partitions in lexicographic order.
%C A194547 Row n has length A000041(n). The sum of row n is equal to zero.
%H A194547 Alois P. Heinz, <a href="/A194547/b194547.txt">Rows n = 1..26, flattened</a>
%F A194547 a(n) = A194546(n) - A193173(n).
%e A194547 Written as a triangle:
%e A194547   0;
%e A194547   -1,1;
%e A194547   -2,0,2;
%e A194547   -3,-1,1,0,3;
%e A194547   -4,-2,0,-1,2,1,4;
%e A194547   -5,-3,-1,-2,1,0,3,-1,2,1,5;
%e A194547   -6,-4,-2,-3,0,-1,2,-2,1,0,4,0,3,2,6;
%e A194547   -7,-5,-3,-4,-1,-2,1,-3,0,-1,3,-1,2,1,5,-2,1,0,4,3,2,7;
%p A194547 T:= proc(n) local b, l;
%p A194547       b:= proc(n, i, t)
%p A194547             if n=0 then l:=l, i-t
%p A194547           elif i>n then
%p A194547           else b(n-i, i, t+1); b(n, i+1, t)
%p A194547             fi
%p A194547           end;
%p A194547       l:= NULL; b(n, 1, 0); l
%p A194547     end:
%p A194547 seq(T(n), n=1..10);  # _Alois P. Heinz_, Dec 22 2011
%t A194547 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];
%t A194547 Table[T[n], {n, 1, 10}] // Flatten (* _Jean-François Alcover_, Mar 05 2021, after _Alois P. Heinz_ *)
%Y A194547 Cf. A105805, A135010, A138121, A194546, A194548, A194549.
%K A194547 sign,tabf,look
%O A194547 1,4
%A A194547 _Omar E. Pol_, Dec 10 2011
%E A194547 More terms from _Alois P. Heinz_, Dec 22 2011