partition -- one more for the collection of proposed problems

[David Ash]

A partition a_i, for i>=1, is a nonincreasing sequence of nonnegative integers for which a_i = 0 for some positive integer i. (Note that this condition implies that all terms beyond a_i are also 0).

The dual, b_i, of a partition a_i is defined as that partition where b_i is the number of elements a_j where a_j >= i.

For example, let 'a' be the following partition:

5, 5, 4, 4, 3, 2, 1, 0, ...

Then the dual, 'b', of 'a' is as follows:

7, 6, 5, 4, 2, 0, ...

Prove that the dual of the dual of a partition is identical to the original partition. A simpler problem would be to simply prove that the dual of a partition is itself a partition.

[t...@bellefleurbooks.com]

Dear David:

This is a good problem. It is easier than the pentagon.

Sincerely,

Tom

On 2021-09-28 20:06, 'David Ash' via Competition Corner Participant

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