On Tue, 1 Jul 2003, Marc LeBrun wrote:
Can you help me get a real citation for Euler's famous partition result (distinct parts = odd parts)?
Sources accessible to me just say he "sent a memoir to the Petersburg Academy" >= 1740, but it'd be nice to be able to give a proper literature reference.
As a matter of fact, it's at my fingertips: Euler, Leonhard. De partitione numerorum. (Latin) Novi commentarii academiae scientiarum Petropolitanae 3 (1750/1), 1753, reprinted in: Commentationes Arithmeticae. (Opera Omnia. Series Prima: Opera Mathematica, Volumen Secundum), 1915, Lipsiae et Berolini, 254--294. Don Knuth gave a talk at MIT about a month ago, about his TAOCP section-in-progress on partitions; the current draft ("pre-fascicle") is available for download at http://www-cs-staff.stanford.edu/~knuth/fasc2d.ps.gz Needless to say, it contains the Euler reference. (The pre-fascicle also has a few exercises that are marked as research problems, including: 59. Is there a Gray path for all partitions of n into powers of 2, where each step either replaces 2^k + 2^k by 2^(k+1) or vice versa? Co-funster Thomas Colthurst and I solved this in the affirmative and found some really interesting combinatorics of the "natural" path -- for example, you can step through the binary partitions of n in this order with each increment taking constant time(!). Our write-up at the moment is at http://people.brandeis.edu/~kleber/binary.pdf in case anyone is interested.) --Michael Kleber kleber@brandeis.edu