Discovering hook length formulas by an expansion technique

Discovering hook length formulas by an expansion technique

We introduce a hook length expansion technique and explain how to discover new hook length formulas for partitions and plane trees by using the Maple package "HookExp". In particular, we derive an expansion formula for the powers of the Euler Product in terms of partition hook lengths, discovered by Nekrasov and Okounkov. We also obtain an extension by adding two more parameters, which appears to be a discrete interpolation between the Macdonald identities and the generating function for t-cores.

The hook length expansion technique and the related formulas are discussed later by Stanley, Ono, Bessenrodt, Carde et al., Panova in the integer partition case; and by Sagan, Chen et al., Yang, Kuba, Eriksen in the plane tree case.