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A234923 Array w(n,h), in which row n shows the weights, as defined in Comments, of the distinct-parts partitions of n, arranged in Mathematica order.

%I A234923 #7 Mar 14 2015 01:26:18
%S A234923 1,2,3,5,4,7,5,9,8,6,11,10,14,7,13,12,11,17,8,15,14,13,20,19,9,17,16,
%T A234923 15,23,14,22,20,10,19,18,17,26,16,25,24,23,30,11,21,20,19,29,18,28,17,
%U A234923 27,26,25,34,12,23,22,21,32,20,31,19,30,29,29,28,38,26
%N A234923 Array w(n,h), in which row n shows the weights, as defined in Comments, of the distinct-parts partitions of n, arranged in Mathematica order.
%C A234923 The weight of a partition P = x (1)+ x(2)+...+x(k) of n is given at A234094 as k*x(1) + (k-1)*x(2) + ... + x(k), which is the number of steps needed to make P from the sum 1+1+...+1 = n by moving dividers into the sum; see the Example section.
%F A234923 w(n,h) = dot product of (partition # h of n) and (k, k-1, ..., 1), where k = length of (partition # h of n).
%e A234923 Represent 1+1+1+1+1 as _1_1_1_1_1_. The partition 3+2+1 matches the placement of dividers d indicated by _1 _1_1d1_1_d_1_d. To place the 1st d takes 3 steps (starting at the 1st '_'); to place the 2nd d takes 3+2 steps (starting at the 1st '_'); to place the 3rd d takes 3+2+1 steps. The total number of steps is 3+5+6 = 14, the 4th number in row 4 because 3+2+1 is the 4th distinct-parts partition of 6 in Mathematica ordering. The first 9 rows:
%e A234923 1
%e A234923 2
%e A234923 3    5
%e A234923 4    7
%e A234923 5    9    8
%e A234923 6   11   10   14
%e A234923 7   13   12   11   17
%e A234923 8   15   14   13   20   19
%e A234923 9   17   16   15   23   14   22   20
%t A234923 p[n_] := p[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &]; 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]]]]]; Flatten[Table[w[n, h], {n, 1, 9}, {h, 1, q[n]}]]
%t A234923 TableForm[Table[w[n, h], {n, 1, 9}, {h, 1, q[n]}]]
%Y A234923 Cf. A234094, A234924.
%K A234923 nonn,easy,tabf
%O A234923 1,2
%A A234923 _Clark Kimberling_, Jan 01 2014