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.

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, 11, 6, 28, 27, 24, 19, 25, 21, 15, 16, 22, 17, 11, 18, 12, 13, 7, 36, 35, 32, 27, 20, 33, 29, 23, 24, 17, 30, 25, 18, 19, 12, 26, 20, 13, 21, 14, 15, 8, 45, 44, 41

Offset: 1

Author

Clark Kimberling, Jan 01 2014

Keywords

• nonn
• easy
• tabf

Comments

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).

Examples

			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:
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   11   6
		

Crossrefs

Cf. A234094, A234097.

Programs

Mathematica

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]]]]];
Flatten[Table[w[n, h], {n, 1, 9}, {h, 1, q[n]}]] (* A234094 *)
TableForm[Table[w[n, h], {n, 1, 9}, {h, 1, q[n]}]]

Formula

w(n,h) = dot product of (partition # h of n) and (k, k-1, ..., 1), where k = length of (partition # h of n).