A350357 Irregular triangle read by rows in which row n lists all elements of the arrangement of the correspondence divisor/part related to the last section of the set of partitions of n in the following order: row n lists the n-th row of A138121 followed by the n-th row of A336812.
1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 3, 1, 4, 2, 2, 1, 1, 1, 1, 2, 4, 1, 2, 1, 5, 3, 2, 1, 1, 1, 1, 1, 1, 5, 1, 3, 1, 2, 1, 1, 6, 3, 3, 4, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 6, 1, 2, 4, 1, 3, 1, 2, 1, 2, 1, 1, 7, 4, 3, 5, 2, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
Offset: 1
Examples
Triangle begins: [1], [1]; [2, 1], [1, 2]; [3, 1, 1], [1, 3, 1]; [4, 2, 2, 1, 1, 1], [1, 2, 4, 1, 2, 1]; [5, 3, 2, 1, 1, 1, 1, 1], [1, 5, 1, 3, 1, 2, 1, 1]; ... Illustration of the first six rows of triangle in an infinite table: |---|---------|-----|-------|---------|-----------|-------------|---------------| | n | | 1 | 2 | 3 | 4 | 5 | 6 | |---|---------|-----|-------|---------|-----------|-------------|---------------| | | | | | | | | 6 | | | | | | | | | 3 3 | | | | | | | | | 4 2 | | P | | | | | | | 2 2 2 | | A | | | | | | 5 | 1 | | R | | | | | | 3 2 | 1 | | T | | | | | 4 | 1 | 1 | | S | | | | | 2 2 | 1 | 1 | | | | | | 3 | 1 | 1 | 1 | | | | | 2 | 1 | 1 | 1 | 1 | | | | 1 | 1 | 1 | 1 | 1 | 1 | |---|---------|-----|-------|---------|-----------|-------------|---------------| | D | A027750 | 1 | 1 2 | 1 3 | 1 2 4 | 1 5 | 1 2 3 6 | | I | A027750 | | | 1 | 1 2 | 1 3 | 1 2 4 | | V | A027750 | | | | 1 | 1 2 | 1 3 | | I | A027750 | | | | | 1 | 1 2 | | S | A027750 | | | | | 1 | 1 2 | | O | A027750 | | | | | | 1 | | R | A027750 | | | | | | 1 | | S | | | | | | | | |---|---------|-----|-------|---------|-----------|-------------|---------------| . For n = 6 in the upper zone of the above table we can see the parts of the last section of the set of partitions of 6 in reverse-colexicographic order in accordance with the 6th row of A138121. In the lower zone of the table we can see the terms from the 6th row of A336812, these are the divisors of the numbers from the 6th row of A336811. Note that in the lower zone of the table every row gives A027750. 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. 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 A336812 and A338156. The growth of the upper zone of the table is in accordance with the growth of the modular prism described in A221529. The growth of the lower zone of the table is in accordance with the growth of the tower described also in A221529. The number of cubic cells added at n-th stage in each polycube is equal to A138879(10) = 150, hence the total number of cubic cells added at n-th stage is equal to 2*A138879(10) = 300, equaling the sum of the 10th row of this triangle.