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A234094 Array {w(n,h)}: row n shows the weights, as defined in Comments, of the partitions of n, arranged in Mathematica order. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..69.

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 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

MATHEMATICA

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]}]]

CROSSREFS

Cf. A234097, A234922, A234924.

Sequence in context: A046530 A003558 A216066 * A301853 A141419 A072451

Adjacent sequences:  A234091 A234092 A234093 * A234095 A234096 A234097

KEYWORD

nonn,easy,tabf

AUTHOR

Clark Kimberling, Jan 01 2014

STATUS

approved

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Last modified February 20 00:28 EST 2019. Contains 320329 sequences. (Running on oeis4.)