Ionuţ Ciocan-Fontanine ; 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 Ciocan-Fontanine 1,1; Matjaž Konvalinka ORCID2; Igor Pak 3

  • 1 School of Mathematics
  • 2 Department of Mathematics, Vanderbilt University
  • 3 Department of Mathematics [UCLA]

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.


Volume: DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010)
Section: Proceedings
Published on: January 1, 2010
Imported on: January 31, 2017
Keywords: Hilbert scheme of points,hook-length formula,bijective proofs,[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO],[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]

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