Based on the ideas in Ciocan-Fontanine, Konvalinka and Pak (2009) , we introduce the weighted analogue of the branching rule for the classical hook length formula, and give two proofs of this result. The first proof is completely bijective, and in a special case gives a new short combinatorial proof of the hook length formula. Our second proof is probabilistic, generalizing the (usual) hook walk proof of Greene, Nijenhuis and Wilf (1979) , as well as the q-walk of Kerov (1993) . Further applications are also presented.
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The authors are grateful to Dennis Stanton for pointing out  to us and explaining its inner working. I.C.-F. would like to thank the Korean Institute for Advanced Studies for support and excellent working conditions. M.K. would like to thank Paul Edelman for several helpful comments on an early draft of this paper. I.P. would like to thank Persi Diaconis for teaching him Kerov’s “segment splitting” algorithm. Partial support for I.C.-F. from the NSF under the grant DMS-0702871 is gratefully acknowledged. I.P. was partially supported by the NSF and the NSA.
- Bijective proof
- Hook-length formula
- Weighted analogue