

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.


2



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
(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 at A234094 as k*x(1) + (k1)*x(2) + ... + x(k).


LINKS



FORMULA

w(n,h) = dot product of (partition # h of n) and (k, k1, ..., 1), where k = length of (partition # h of n).


EXAMPLE

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


MATHEMATICA

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]}]]


CROSSREFS



KEYWORD

nonn,easy,tabf


AUTHOR



STATUS

approved



