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A234922 Array w(n,h), in which row n shows the weights (defined in Comments) of the partitions of n, arranged in reverse Mathematica order.

%I A234922 #10 Mar 14 2015 01:25:31
%S A234922 1,3,2,6,5,3,10,9,6,7,4,15,14,11,12,8,9,5,21,20,17,12,18,14,9,15,10,
%T A234922 11,6,28,27,24,19,25,21,15,16,22,17,11,18,12,13,7,36,35,32,27,20,33,
%U A234922 29,23,24,17,30,25,18,19,12,26,20,13,21,14,15,8,45,44,41
%N A234922 Array w(n,h), in which row n shows the weights (defined in Comments) of the partitions of n, arranged in reverse Mathematica order.
%C A234922 The weight of a partition P = x(1)+ x(2)+...+x(k) of n is introduced at A234094 as k*x(1) + (k-1)*x(2) + ... + x(k).
%F A234922 w(n,h) = dot product of (partition # h of n) and (k, k-1, ..., 1), where k = length of (partition # h of n).
%e A234922 Represent 1+1+1+1+1 as _1_1_1_1_1_.  The partition 1+2+2 matches the placement of dividers d indicated by _1d1_1d1_1d. To place the 1st d takes 1 step (starting at the 1st '_'); to place the 2nd d takes 1+2 steps (starting at the 1st '_'); to place the 3rd d takes 1+2+2 steps. The total number of steps is 2+3+5 = 9, the 3rd number in row 5, because 1+2+2 is the 3rd partition of 5 in reverse Mathematica ordering. The first 6 rows:
%e A234922 1
%e A234922 3    2
%e A234922 6    5    3
%e A234922 10   9    6    7    4
%e A234922 15   14   11   12   8    9    5
%e A234922 21   20   17   12   18   14   9   15   10   11   6
%t A234922 p[n_] := p[n] = Reverse[IntegerPartitions[n]]; q[n_] := q[n] = Length[p[n]]; v[n_] := v[n] = Table[n + 1 - i, {i, 1, n}]; w[n_, h_] := w[n, h] = Dot[p[n][[h]], v[Length[p[n][[h]]]]];
%t A234922 Flatten[Table[w[n, h], {n, 1, 9}, {h, 1, q[n]}]] (* A234094 *)
%t A234922 TableForm[Table[w[n, h], {n, 1, 9}, {h, 1, q[n]}]]
%Y A234922 Cf. A234094, A234097.
%K A234922 nonn,easy,tabf
%O A234922 1,2
%A A234922 _Clark Kimberling_, Jan 01 2014