cp's OEIS Frontend

A350333 Irregular triangle read by rows in which row n lists all elements of the arrangement of the correspondence divisor/part related to the partitions of n in the following order: row n lists the n-th row of A026792 followed by the n-th row of A338156.

%I A350333 #67 Jan 04 2022 02:55:42
%S A350333 1,1,2,1,1,1,2,1,3,2,1,1,1,1,1,3,1,2,1,1,4,2,2,3,1,2,1,1,1,1,1,1,1,2,
%T A350333 4,1,3,1,2,1,2,1,1,1,5,3,2,4,1,2,2,1,3,1,1,2,1,1,1,1,1,1,1,1,1,5,1,2,
%U A350333 4,1,3,1,3,1,2,1,2,1,2,1,1,1,1,1,6,3,3,4,2,2,2,2,5,1
%N A350333 Irregular triangle read by rows in which row n lists all elements of the arrangement of the correspondence divisor/part related to the partitions of n in the following order: row n lists the n-th row of A026792 followed by the n-th row of A338156.
%e A350333 Triangle begins:
%e A350333 [1], [1];
%e A350333 [2, 1, 1], [1, 2, 1];
%e A350333 [3, 2, 1, 1, 1, 1], [1, 3, 1, 2, 1, 1];
%e A350333 [4, 2, 2, 3, 1, 2, 1, 1, 1, 1, 1, 1], [1, 2, 4, 1, 3, 1, 2, 1, 2, 1, 1, 1];
%e A350333 ...
%e A350333 Illustration of the first six rows of triangle in an infinite table:
%e A350333 .
%e A350333 |---|---------|-----|-------|---------|-----------|-------------|---------------|
%e A350333 | n |         |  1  |   2   |    3    |     4     |      5      |       6       |
%e A350333 |---|---------|-----|-------|---------|-----------|-------------|---------------|
%e A350333 |   |         |     |       |         |           |             |  6            |
%e A350333 | P |         |     |       |         |           |             |  3 3          |
%e A350333 | A |         |     |       |         |           |             |  4 2          |
%e A350333 | R |         |     |       |         |           |             |  2 2 2        |
%e A350333 | T |         |     |       |         |           |  5          |  5 1          |
%e A350333 | I |         |     |       |         |           |  3 2        |  3 2 1        |
%e A350333 | T |         |     |       |         |  4        |  4 1        |  4 1 1        |
%e A350333 | I |         |     |       |         |  2 2      |  2 2 1      |  2 2 1 1      |
%e A350333 | O |         |     |       |  3      |  3 1      |  3 1 1      |  3 1 1 1      |
%e A350333 | N |         |     |  2    |  2 1    |  2 1 1    |  2 1 1 1    |  2 1 1 1 1    |
%e A350333 | S |         |  1  |  1 1  |  1 1 1  |  1 1 1 1  |  1 1 1 1 1  |  1 1 1 1 1 1  |
%e A350333 ----|---------|-----|-------|---------|-----------|-------------|---------------|
%e A350333 |   | A027750 |  1  |  1 2  |  1   3  |  1 2   4  |  1       5  |  1 2 3     6  |
%e A350333 |   | A027750 |     |  1    |  1 2    |  1   3    |  1 2   4    |  1       5    |
%e A350333 |   | A027750 |     |       |  1      |  1 2      |  1   3      |  1 2   4      |
%e A350333 |   | A027750 |     |       |  1      |  1 2      |  1   3      |  1 2   4      |
%e A350333 |   | A027750 |     |       |         |  1        |  1 2        |  1   3        |
%e A350333 | D | A027750 |     |       |         |  1        |  1 2        |  1   3        |
%e A350333 | I | A027750 |     |       |         |  1        |  1 2        |  1   3        |
%e A350333 | V | A027750 |     |       |         |           |  1          |  1 2          |
%e A350333 | I | A027750 |     |       |         |           |  1          |  1 2          |
%e A350333 | S | A027750 |     |       |         |           |  1          |  1 2          |
%e A350333 | O | A027750 |     |       |         |           |  1          |  1 2          |
%e A350333 | R | A027750 |     |       |         |           |  1          |  1 2          |
%e A350333 | S | A027750 |     |       |         |           |             |  1            |
%e A350333 |   | A027750 |     |       |         |           |             |  1            |
%e A350333 |   | A027750 |     |       |         |           |             |  1            |
%e A350333 |   | A027750 |     |       |         |           |             |  1            |
%e A350333 |   | A027750 |     |       |         |           |             |  1            |
%e A350333 |   | A027750 |     |       |         |           |             |  1            |
%e A350333 |   | A027750 |     |       |         |           |             |  1            |
%e A350333 |---|---------|-----|-------|---------|-----------|-------------|---------------|
%e A350333 .
%e A350333 For n = 6 in the upper zone of the above table we can see the partitions of 6 in reverse-colexicographic order in accordance with the 6th row of A026792.
%e A350333 In the lower zone of the table we can see the terms from the 6th row of A338156, these are the divisors of the numbers from the 6th row of A176206.
%e A350333 Note that in the lower zone of the table every row gives A027750.
%e A350333 The total number of rows in the table is equal to A000070(6+1) = 30.
%e A350333 The remarkable fact is that the elements in the lower zone of the arrangement are the same as the elements in the upper zone but in other order.
%e A350333 For an explanation of the connection of the elements of the upper zone with the elements of the lower zone, that is the correspondence divisor/part, see A338156.
%e A350333 For n = 10 we can see a representation of the upper zone (the partitions) and of the lower zone (the divisors) with the two polycubes described in A221529 respectively: a prism of partitions and a tower whose terraces are the symmetric representation of sigma(m), for m = 1..10. Each polycube has A066186(10) = 420 cubic cells, hence the total number of cubic cells is equal to A220909(10) = 840, equaling the sum of the 10th row of this triangle.
%Y A350333 Row sums give A220909.
%Y A350333 Row lengths give A211978.
%Y A350333 Cf. A350357 (analog for the last section of the set of partitions of n).
%Y A350333 Cf. A000041, A000070, A006128, A026792, A027750, A066186, A135010, A176206, A221529, A237593, A336811, A336812, A338156, A340035.
%K A350333 nonn,tabf
%O A350333 1,3
%A A350333 _Omar E. Pol_, Dec 25 2021