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A341149 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.

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