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.

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, 15, 23, 14, 22, 20, 10, 19, 18, 17, 26, 16, 25, 24, 23, 30, 11, 21, 20, 19, 29, 18, 28, 17, 27, 26, 25, 34, 12, 23, 22, 21, 32, 20, 31, 19, 30, 29, 29, 28, 38, 26

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 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.

Examples

			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:
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   15   23   14   22   20
		

Crossrefs

Cf. A234094, A234924.

Programs

Mathematica

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]}]]
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).