Question

Recall the permutation of the partitions of n given by taking the transpose of the dot diagram. Find all the partitions that are sent by this permutation to themselves for n=4,5 and n=10.

Answer 1

For n=4, the partitions that are sent to themselves are (4), (3, 1), (2, 2), (2, 1, 1) and (1, 1, 1, 1). For n=5, the partitions that are sent to themselves are (5), (4, 1), (3, 2), (3, 1, 1), (2, 2, 1), (2, 1, 1, 1) and (1, 1, 1, 1, 1). For n=10, the partitions that are sent to themselves are (10), (9,1), (8,2), (7,3), (7,2,1), (6,4), (6,3,1), (6,2,2), (5,5), (5,4,1), (5,3,2), (4,4,2), (4,3,3), (4,3,2,1), (4,2,2,2), (3,3,3,1), (3,3,2,2), (3,2,2,2,1) and (2,2,2,2,2).

Explanation:

The permutation of the partitions of n is found by taking the transpose of the dot diagram. A dot diagram is a graphical representation of a partition. It is a rectangular diagram with n dots in n rows. Each row represents a partition. The columns represent the parts in the partition. For example, a partition of 4 would be (4), (3,1), (2,2), (2,1,1) and (1,1,1,1). Each partition can be represented in a dot diagram as 4 dots in 4 rows. When the dot diagram is transposed, it will represent the permutation of the given partition.

The number of partitions of n is given by the number of ways to partition n into a sum of positive integers. For example, to partition n=4, we can have 4 ways: (4), (3,1), (2,2) and (1,1,1,1). This can be represented in the dot diagram as 4 dots in 4 rows. When the dot diagram is transposed, it will represent the permutation of the given partition.

For n=4, the partitions that are sent to themselves are (4), (3, 1), (2, 2), (2, 1, 1) and (1, 1, 1, 1). This can be determined by looking at the transposed dot diagram and checking which partitions are the same as the original.

For n=5, the partitions that are sent to themselves are (5), (4, 1), (3, 2), (3, 1, 1), (2, 2, 1), (2, 1, 1, 1) and (1, 1, 1, 1, 1). This can also be determined by looking at the transposed dot diagram and checking which partitions are the same as the original.

Therefore, for n=4,5 and 10, the partitions that are sent to themselves can be determined by looking at the transposed dot diagram and checking which partitions are the same as the original.