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Hook Length Formulas for Partitions and Plane Trees
Some conjectures and open problems on partition hook lengths
By Guo-Niu Han
[download the paper: .ps -
.pdf, 15 pages, 2008/05/11]
Abstract.
We present some conjectures and open problems on partition hook lengths,
which are all motivated by known results on the subject.
The conjectures are suggested by extensive experimental calculations using
a computer algebra system.
The first conjecture unifies two classical results on the number of
standard Young tableaux and the number of pairs of standard Young tableaux
of the same shape. The second unifies the classical hook formula and
the marked hook formula.
The third implies the long standing Lehmer conjecture which says that
the Ramanujan tau-function never takes the zero value.
The fourth is a more
precise version of the third one in the case of 3-cores.
We also list some open problems on partition hook lengths.
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Last update: 2008/05/11