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

1,2

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.

Links

Table of n, a(n) for n=1..68.

Formula

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

Example

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

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]}]]

Crossrefs

Cf. A234094, A234924.

Sequence in context: A085180 A360209 A114750 * A145391 A371479 A057756

Adjacent sequences: A234920 A234921 A234922 * A234924 A234925 A234926

Keyword

nonn,easy,tabf

Author

Clark Kimberling, Jan 01 2014

Status

approved