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%I #64 Jan 20 2023 01:31:29
%S 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,
%T 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,
%U 1,1,1,1,1,1,1,1,1,1,1,1,1,1,11,7,7,7,5,5,5,5,3,3,3,3,3,3,3
%N Irregular triangle read by rows T(n,k) in which row n lists n blocks where the m-th block consists of A000203(m) copies of A000041(n-m), with 1 <= m <= n.
%C 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).
%H Paolo Xausa, <a href="/A341149/b341149.txt">Table of n, a(n) for n = 1..12451</a> (rows 1..35 of triangle, flattened)
%e Triangle begins:
%e 1;
%e 1,1,1,1;
%e 2,1,1,1,1,1,1,1;
%e 3,2,2,2,1,1,1,1,1,1,1,1,1,1,1;
%e 5,3,3,3,2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1;
%e 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;
%e ...
%e For n = 6 we have that:
%e Row 6 Row 6 of
%e m A000203(m) A000041(n-m) block(m) A221529
%e 1 1 7 [7] 7
%e 2 3 5 [5,5,5] 15
%e 3 4 3 [3,3,3,3] 12
%e 4 7 2 [2,2,2,2,2,2,2] 14
%e 5 6 1 [1,1,1,1,1,1] 6
%e 6 12 1 [1,1,1,1,1,1,1,1,1,1,1,1] 12
%e .
%e 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.
%e 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.
%e _ _ _ _ _ _
%e 11 |_ _ _ | 6
%e |_ _ _|_ | 3 3
%e |_ _ | | 4 2
%e |_ _|_ _|_ | 2 2 2 _
%e 7 |_ _ _ | | 5 1 | |
%e |_ _ _|_ | | 3 2 1 |_|_
%e 5 |_ _ | | | 4 1 1 | |
%e |_ _|_ | | | 2 2 1 1 |_ _|_
%e 3 |_ _ | | | | 3 1 1 1 |_ _|_|_
%e 2 |_ | | | | | 2 1 1 1 1 |_ _ _|_|_ _
%e 1 |_|_|_|_|_|_| 1 1 1 1 1 1 |_ _ _ _|_|_|
%e .
%e Figure 1. Figure 2. Figure 3.
%e Front view Partitions Lateral view
%e of the prism of 6. of the tower.
%e of partitions.
%e .
%e Row 6 of
%e _ _ _ _ _ _ A341148
%e 1 |_| | | | | 7 5 3 2 1 1 19
%e 2 |_ _|_| | | 5 5 3 2 1 1 17
%e 3 |_ _| _| | 3 3 2 2 1 1 12
%e 4 |_ _ _| _| 2 2 2 1 1 1 9
%e 5 | _| 1 1 1 1 1 5
%e 6 |_ _ _ _| 1 1 1 1 4
%e .
%e Figure 4. Figure 5.
%e Top view Heights
%e of the tower. in the
%e top view.
%e .
%e 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.
%e 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].
%t A341149row[n_]:=Flatten[Array[ConstantArray[PartitionsP[n-#],DivisorSigma[1,#]]&,n]];
%t nrows=7;Array[A341149row,nrows] (* _Paolo Xausa_, Jun 20 2022 *)
%Y Every column gives A000041.
%Y Row lengths give A024916.
%Y Row sums give the nonzero terms of A066186.
%Y Cf. A000203, A221529, A236104, A237270, A237271, A237593, A337209, A339106, A340584, A341148, A345023.
%K nonn,tabf
%O 1,6
%A _Omar E. Pol_, Feb 06 2021