Ionuţ CiocanFontanine ; Matjaž Konvalinka ; Igor Pak

Weighted branching formulas for the hook lengths
dmtcs:2850 
Discrete Mathematics & Theoretical Computer Science,
January 1, 2010,
DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010)

https://doi.org/10.46298/dmtcs.2850
Weighted branching formulas for the hook lengthsArticle
Authors: Ionuţ CiocanFontanine ^{1,}^{1}; Matjaž Konvalinka ^{2}; Igor Pak ^{3}
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Ionuţ CiocanFontanine;Matjaž Konvalinka;Igor Pak
1 School of Mathematics
2 Department of Mathematics, Vanderbilt University
3 Department of Mathematics [UCLA]
The famous hooklength formula is a simple consequence of the branching rule for the hook lengths. While the GreeneNijenhuisWilf 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.