A234094 -id:A234094 - cp's OEIS frontend

A234097 Combined weight, as defined at A234094, of the partitions of n.

1, 5, 14, 36, 74, 153, 275, 497, 832, 1383, 2182, 3446, 5213, 7872, 11563, 16899, 24155, 34422, 48146, 67117, 92239, 126222, 170721, 230113, 306990, 408110, 538067, 706834, 921862, 1198453, 1548054, 1993462, 2552970, 3259507, 4141423, 5247231, 6618965

Offset: 1

Clark Kimberling, Jan 01 2014

These are the row sums of the array at A234094, where weight is defined.

Cf. A234094, A234922, A234924.

z = 40; p[n_] := p[n] = IntegerPartitions[n]; q[n_] := q[n] = Length[p[n]]; v[n_] := v[n]; w[n_, h_] := w[n, h] = Dot[p[n][[h]], v[Length[p[n][[h]]]]]; a[n_] := Sum[w[n, h], {h, 1, q[n]}]; t = Table[a[n], {n, 1, z}]

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.

Original entry on oeis.org

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

Clark Kimberling, Jan 01 2014

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).

			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

Cf. A234094, A234097.

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]}]]

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

Original entry on oeis.org

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

Clark Kimberling, Jan 01 2014

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.

			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

Cf. A234094, A234924.

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]}]]

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

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A234097 Combined weight, as defined at A234094, of the partitions of n.

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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.

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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.

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Mathematica

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