## Some conjectures and open problems on partition hook lengths.(English)Zbl 1167.05004

Summary: We present some conjectures and open problems on partition hook lengths motivated by known results on the subject. The conjectures were 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 includes the longstanding Lehmer conjecture, which says that the Ramanujan tau function never assumes the value zero. The fourth is a more precise version of the third in the case of 3-cores. We also list some open problems on partition hook lengths.

### MSC:

 05A15 Exact enumeration problems, generating functions 05A17 Combinatorial aspects of partitions of integers 05A19 Combinatorial identities, bijective combinatorics 05E10 Combinatorial aspects of representation theory 11D45 Counting solutions of Diophantine equations 11P81 Elementary theory of partitions

### Keywords:

partitions; hook length formulas; Lehmer conjecture; $$t$$-cores

HookExp
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