Odd & Distinct Partitions

I was reviewing my Anki flashcards from my second year discrete maths course (MATH2302) and rediscovered this fact for myself: "For any integer, the number of odd partitions is the same as the number of distinct partitions." I found this strange; it was clear from the generating functions, but I didn't have any intuition on why this would be the case.

To give some background, a partition is a way to write an integer as a sum of positive integers where we ignore ordering ...