Nice mapping between odd partitions and distinct partitions. Reminds me of the slick mapping between distinct odd partitions and self-dual partitions. Bend the odd numbers into gnomons and nest them: Distinct odd partition (7, 5, 1) A A AB AB AB AB ABC maps to self-dual partition (4, 4, 3, 2) AB ABC ABBB AAAA ----- Original Message ----- From: Allan C. Wechsler To: math-fun Sent: Thursday, May 15, 2003 3:45 PM Subject: Re: [math-fun] distinct and square-free factorizations [omitted stuff] The combinatorial proof shows a direct 1-1 correspondence between partitions of the two types. To convert a partition of the "odd" type to one of the "distinct" type, find a pair of identical piles and merge them. Repeat until there are no identical piles; by construction the resulting partition is of the "distinct" type. The inverse operation is to find an even pile and break it into two equal ones. This operation is repeated until there are no even piles, so the result is a partition of "odd" type. [more omitted stuff] -A