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Weighted branching formulas for the hook lengths

Abstract : The famous hook-length formula is a simple consequence of the branching rule for the hook lengths. While the Greene-Nijenhuis-Wilf probabilistic proof is the most famous proof of the rule, it is not completely combinatorial, and a simple bijection was an open problem for a long time. In this extended abstract, we show an elegant bijective argument that proves a stronger, weighted analogue of the branching rule. Variants of the bijection prove seven other interesting formulas. Another important approach to the formulas is via weighted hook walks; we discuss some results in this area. We present another motivation for our work: $J$-functions of the Hilbert scheme of points.
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Ionuţ Ciocan-Fontanine, Matjaž Konvalinka, Igor Pak. Weighted branching formulas for the hook lengths. 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010), 2010, San Francisco, United States. pp.259-270. ⟨hal-01186278⟩

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