MATH 310 Final: MATH 310 Amherst F16M310Final

• School: Amherst College
• Department: Mathematics and Statistics
• Course: MATH 310
• Professor: All

Document Summary

Math 310 intro. to the theory of partitions. If |s| = , assume |q| < 1, and |qz| < 1. Xm=0 p(n | m distinct parts in s)zmqn = yn s. Remark. p(n | m distinct parts in s) may also be written as p(cid:16)n(cid:12)(cid:12)(cid:12) = p(n | odd number of parts) + p(n | largest part is even). Prove for any n n0 that p(n) = p (n). In addition, explicitly verify this result for n = 10 by computing and counting partitions. Complete any one of the three problems a1, a2, a3 listed below. (a1) prove for n n0 that n. =(cid:20) 2n n (cid:21) . (a2) prove the pentagonal number theorem. That is, prove for any n n that p(n | even # distinct parts) = p(n | odd # distinct parts) + e(n), where e(n) =(( 1)j,