A direct bijective proof of the hook-length formula

Jean-Christophe Novelli; Igor Pak; Alexander V. Stoyanovskii

doi:10.46298/dmtcs.239

Jean-Christophe Novelli ; Igor Pak ; Alexander V. Stoyanovskii - A direct bijective proof of the hook-length formula

dmtcs:239 - Discrete Mathematics & Theoretical Computer Science, January 1, 1997, Vol. 1 - https://doi.org/10.46298/dmtcs.239

A direct bijective proof of the hook-length formula

Authors: Jean-Christophe Novelli ¹; Igor Pak ²; Alexander V. Stoyanovskii ³

1 Laboratoire d'informatique Algorithmique : Fondements et Applications
2 Department of Mathematics [Cambridge]
3 Department of Mathematics [Moscou]

This paper presents a new proof of the hook-length formula, which computes the number of standard Young tableaux of a given shape. After recalling the basic definitions, we present two inverse algorithms giving the desired bijection. The next part of the paper presents the proof of the bijectivity of our construction. The paper concludes with some examples.

https://doi.org/10.46298/dmtcs.239

Source: HAL:hal-00955690v1

Volume: Vol. 1

Published on: January 1, 1997

Imported on: March 26, 2015

Keywords: Hook-length formula,bijective proof,inverse algorithms,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]

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Source : ScholeXplorer IsRelatedTo DOI 10.1016/0001-8708(79)90023-9 10.1016/0001-8708(79)90023-9 A probabilistic proof of a formula for the number of Young tableaux of a given shape

12 Documents citing this article

Source : OpenCitations

2016, Mean And Insensitive. Random Permutations, Combinatorics Of Permutations, pp. 253-292, 10.1201/b12210-13.

Bodini, Olivier; Fusy, Ăric; Pivoteau, Carine, 2009, Random Sampling Of Plane Partitions, Combinatorics, Probability And Computing, 19, 2, pp. 201-226, 10.1017/s0963548309990332.

Ciocan-Fontanine, IonuĹŁ; Konvalinka, MatjaĹž; Pak, Igor, 2011, The Weighted Hook Length Formula, Journal Of Combinatorial Theory, Series A, 118, 6, pp. 1703-1717, 10.1016/j.jcta.2011.02.004.

Ciocan-Fontanine, IonuĹŁ; Konvalinka, MatjaĹž; Pak, Igor, 2012, Quantum Cohomology Of Hilbn(C2) And The Weighted Hook Walk On Young Diagrams, Journal Of Algebra, 349, 1, pp. 268-283, 10.1016/j.jalgebra.2011.10.011.

Griffiths, Martin; Lord, Nick, 2011, The Hook-Length Formula And Generalised Catalan Numbers, The Mathematical Gazette, 95, 532, pp. 23-30, 10.1017/s002555720000228x.

Konvalinka, MatjaĹž, 2020, A Bijective Proof Of The Hook-Length Formula For Skew Shapes, European Journal Of Combinatorics, 88, pp. 103104, 10.1016/j.ejc.2020.103104.

Konvalinka, MatjaĹž, 2017, A Bijective Proof Of The Hook-Length Formula For Skew Shapes, Electronic Notes In Discrete Mathematics, 61, pp. 765-771, 10.1016/j.endm.2017.07.034.

Morales, A. H.; Pak, I.; Panova, G., 2022, Hook Formulas For Skew Shapes IV. Increasing Tableaux And Factorial Grothendieck Polynomials, Journal Of Mathematical Sciences, 261, 5, pp. 630-657, 10.1007/s10958-022-05777-0.

Morales, Alejandro H.; Pak, Igor; Panova, Greta, 2018, Hook Formulas For Skew Shapes I. Q-Analogues And Bijections, Journal Of Combinatorial Theory, Series A, 154, pp. 350-405, 10.1016/j.jcta.2017.09.002.

Neumann, Christoph; Sulzgruber, Robin, 2015, A Complexity Theorem For The NovelliâPakâStoyanovskii Algorithm, Journal Of Combinatorial Theory, Series A, 135, pp. 85-104, 10.1016/j.jcta.2015.04.001.

Sun, Brian Y.; Hu, Yingying, 2018, A Probabilistic Method For The Number Of Standard Immaculate Tableaux, Rocky Mountain Journal Of Mathematics, 48, 6, 10.1216/rmj-2018-48-6-2087.

Zhang, Rong, 2010, On An Identity Of Glass And Ng Concerning The Hook Length Formula, Discrete Mathematics, 310, 17-18, pp. 2440-2442, 10.1016/j.disc.2010.05.013.

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