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Discrete Mathematics & Theoretical Computer Science |
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
2; 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.