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A234094
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Array {w(n,h)}: row n shows the weights, as defined in Comments, of the partitions of n, arranged in Mathematica order.
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4
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1, 2, 3, 3, 5, 6, 4, 7, 6, 9, 10, 5, 9, 8, 12, 11, 14, 15, 6, 11, 10, 15, 9, 14, 18, 12, 17, 20, 21, 7, 13, 12, 18, 11, 17, 22, 16, 15, 21, 25, 19, 24, 27, 28, 8, 15, 14, 21, 13, 20, 26, 12, 19, 18, 25, 30, 17, 24, 23, 29, 33, 20, 27, 32, 35, 36, 9, 17, 16
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OFFSET
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1,2
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COMMENTS
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The weight of a partition P = x(1)+x(2)+...+x(k) of n is introduced here 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 (or parentheses) into the sum; see the Example section.
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LINKS
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FORMULA
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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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EXAMPLE
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Represent 1+1+1+1+1 as _1_1_1_1_1_. The partition 2+2+1 matches the placement of dividers d indicated by _1_1d1_1d1d. To place the 1st d takes 2 steps (starting at the 1st _); to place the 2nd d takes 2+2 = 4 steps (starting at the 1st _ ); to place the 3rd d takes 2+2+1 = 5 steps. The total number of steps is 2+4+5 = 11, which is the 5th number in row 5 because 2+2+1 is the 5th partition of 5 in Mathematica ordering. The first 6 rows are:
1
2 ... 3
3 ... 5 ... 6
4 ... 7 ... 6 ... 9 ... 10
5 ... 9 ... 8 ... 12 .. 11 .. 14 ... 15
6 ... 11 .. 10 .. 15 .. 9 ... 14 ... 18 .. 12 .. 17 .. 20 .. 21
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MATHEMATICA
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p[n_] := p[n] = 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]}]]
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CROSSREFS
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KEYWORD
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nonn,easy,tabf
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AUTHOR
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STATUS
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approved
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