The weighted hook length formula

Authors:
Ionu Ciocan-Fontanine;Matja Konvalinka;Igor Pak
Affiliations:
School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA;Department of Mathematics, Vanderbilt University, Nashville, TN 37240, USA;Department of Mathematics, UCLA, Los Angeles, CA 90095, USA
Venue:
Journal of Combinatorial Theory Series A
Year:
2011

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Abstract

Based on the ideas in Ciocan-Fontanine, Konvalinka and Pak (2009) [5], 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) [15], as well as the q-walk of Kerov (1993) [20]. Further applications are also presented.