Let N be an integer.Is number of ways to express N as the sum of odd parts is equal to the number of ways to express n as the sum of distinct parts ?
When you say odd parts do you mean integers as odd or number of parts as odd ?
In 1748 Euler proved that for any positive integer n the number of partitions of n into odd parts is the same as the number of partitions of n into distinct parts.There are well known proofs of this identity. One of them is by Sylvester, which can be found on pages 28–30 in:G. E. Andrews. J. J. Sylvester, Johns Hopkins and Partitions. In A Century of Mathematics in America, Part I, pages 21–40. American Mathematical Society, 1988, and on pages 13–14 in: Percy A. MacMahon, Combinatory Analysis, Volume II. Chelsea Publishing Company, New York, N.Y., 1984.
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