Some Combinatorial Properties of Hook Lengths, Contents, and Parts of Partitions

Richard P. Stanley

Published 2008-07-02, updated 2009-04-10Version 3

This paper proves a generalization of a conjecture of Guoniu Han, inspired originally by an identity of Nekrasov and Okounkov. The main result states that certain sums over partitions p of n, involving symmetric functions of the squares of the hook lengths of p, are polynomial functions of n. A similar result is obtained for symmetric functions of the contents and shifted parts of n.