OFFSET
1,6
COMMENTS
In the n-th row of the triangle the values of the m-th block are the number of cubes that are exactly below every cell of the symmetric representation of sigma(m) in the tower described in A221529 (see figure 5 in the example here).
LINKS
Paolo Xausa, Table of n, a(n) for n = 1..12451 (rows 1..35 of triangle, flattened)
EXAMPLE
Triangle begins:
1;
1,1,1,1;
2,1,1,1,1,1,1,1;
3,2,2,2,1,1,1,1,1,1,1,1,1,1,1;
5,3,3,3,2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1;
7,5,5,5,3,3,3,3,2,2,2,2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1;
...
For n = 6 we have that:
Row 6 Row 6 of
1 1 7 [7] 7
2 3 5 [5,5,5] 15
3 4 3 [3,3,3,3] 12
4 7 2 [2,2,2,2,2,2,2] 14
5 6 1 [1,1,1,1,1,1] 6
6 12 1 [1,1,1,1,1,1,1,1,1,1,1,1] 12
.
so the 6th row of triangle is [7,5,5,5,3,3,3,3,2,2,2,2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] and the row sums equals A066186(6) = 66.
We can see below some views of two associated polycubes called "prism of partitions" and "tower". Both objects contains the same number of cubes (that property is also valid for n >= 1). For further information about these two associated objects see A221529.
_ _ _ _ _ _
11 |_ _ _ | 6
|_ _ _|_ | 3 3
|_ _ | | 4 2
|_ _|_ _|_ | 2 2 2 _
7 |_ _ _ | | 5 1 | |
|_ _ _|_ | | 3 2 1 |_|_
5 |_ _ | | | 4 1 1 | |
|_ _|_ | | | 2 2 1 1 |_ _|_
3 |_ _ | | | | 3 1 1 1 |_ _|_|_
2 |_ | | | | | 2 1 1 1 1 |_ _ _|_|_ _
1 |_|_|_|_|_|_| 1 1 1 1 1 1 |_ _ _ _|_|_|
.
Figure 1. Figure 2. Figure 3.
Front view Partitions Lateral view
of the prism of 6. of the tower.
of partitions.
.
Row 6 of
_ _ _ _ _ _ A341148
1 |_| | | | | 7 5 3 2 1 1 19
2 |_ _|_| | | 5 5 3 2 1 1 17
3 |_ _| _| | 3 3 2 2 1 1 12
4 |_ _ _| _| 2 2 2 1 1 1 9
5 | _| 1 1 1 1 1 5
6 |_ _ _ _| 1 1 1 1 4
.
Figure 4. Figure 5.
Top view Heights
of the tower. in the
top view.
.
Figure 5 shows the heights of the terraces of the tower, or in other words the number of cubes in the column exactly below every cell of the top view. For example: in the 6th row of triangle the first block is [7] because there are seven cubes exactly below the symmetric representation of sigma(1) = 1. The second block is [5, 5, 5] because there are five cubes exactly below every cell of the symmetric representation of sigma(2) = 3. The third block is [3, 3, 3, 3] because there are three cubes exactly below every cell of the symmetric representation of sigma(3) = 4, and so on.
Note that the terraces that are the symmetric representation of sigma(5) and the terraces that are the symmetric representation of sigma(6) both are unified in level 1 of the structure. That is because the first two partition numbers A000041 are [1, 1].
MATHEMATICA
A341149row[n_]:=Flatten[Array[ConstantArray[PartitionsP[n-#], DivisorSigma[1, #]]&, n]];
nrows=7; Array[A341149row, nrows] (* Paolo Xausa, Jun 20 2022 *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Omar E. Pol, Feb 06 2021
STATUS
approved