Some Conjectures and Open Problems on Partition Hook Lengths

Abstract

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.