The weighted hook length formula

Ionuţ Ciocan-Fontanine; Matjaž Konvalinka; Igor Pak

doi:10.1016/j.jcta.2011.02.004

The weighted hook length formula

Ionuţ Ciocan-Fontanine, Matjaž Konvalinka, Igor Pak

School of Mathematics

Research output: Contribution to journal › Article › peer-review

13 Scopus citations

Abstract

Based on the ideas in Ciocan-Fontanine, Konvalinka and Pak (2009) [5], we introduce the weighted analogue of the branching rule for the classical hook length formula, and give two proofs of this result. The first proof is completely bijective, and in a special case gives a new short combinatorial proof of the hook length formula. Our second proof is probabilistic, generalizing the (usual) hook walk proof of Greene, Nijenhuis and Wilf (1979) [15], as well as the q-walk of Kerov (1993) [20]. Further applications are also presented.

Original language	English (US)
Pages (from-to)	1703-1717
Number of pages	15
Journal	Journal of Combinatorial Theory. Series A
Volume	118
Issue number	6
DOIs	https://doi.org/10.1016/j.jcta.2011.02.004
State	Published - Aug 2011

Bibliographical note

Funding Information:
The authors are grateful to Dennis Stanton for pointing out [4] to us and explaining its inner working. I.C.-F. would like to thank the Korean Institute for Advanced Studies for support and excellent working conditions. M.K. would like to thank Paul Edelman for several helpful comments on an early draft of this paper. I.P. would like to thank Persi Diaconis for teaching him Kerov’s “segment splitting” algorithm. Partial support for I.C.-F. from the NSF under the grant DMS-0702871 is gratefully acknowledged. I.P. was partially supported by the NSF and the NSA.

Keywords

Bijective proof
Hook-length formula
Weighted analogue

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10.1016/j.jcta.2011.02.004

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title = "The weighted hook length formula",

abstract = "Based on the ideas in Ciocan-Fontanine, Konvalinka and Pak (2009) [5], we introduce the weighted analogue of the branching rule for the classical hook length formula, and give two proofs of this result. The first proof is completely bijective, and in a special case gives a new short combinatorial proof of the hook length formula. Our second proof is probabilistic, generalizing the (usual) hook walk proof of Greene, Nijenhuis and Wilf (1979) [15], as well as the q-walk of Kerov (1993) [20]. Further applications are also presented.",

keywords = "Bijective proof, Hook-length formula, Weighted analogue",

author = "Ionu{\c t} Ciocan-Fontanine and Matja{\v z} Konvalinka and Igor Pak",

note = "Funding Information: The authors are grateful to Dennis Stanton for pointing out [4] to us and explaining its inner working. I.C.-F. would like to thank the Korean Institute for Advanced Studies for support and excellent working conditions. M.K. would like to thank Paul Edelman for several helpful comments on an early draft of this paper. I.P. would like to thank Persi Diaconis for teaching him Kerov{\textquoteright}s “segment splitting” algorithm. Partial support for I.C.-F. from the NSF under the grant DMS-0702871 is gratefully acknowledged. I.P. was partially supported by the NSF and the NSA.",

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AU - Ciocan-Fontanine, Ionuţ

AU - Konvalinka, Matjaž

AU - Pak, Igor

N1 - Funding Information: The authors are grateful to Dennis Stanton for pointing out [4] to us and explaining its inner working. I.C.-F. would like to thank the Korean Institute for Advanced Studies for support and excellent working conditions. M.K. would like to thank Paul Edelman for several helpful comments on an early draft of this paper. I.P. would like to thank Persi Diaconis for teaching him Kerov’s “segment splitting” algorithm. Partial support for I.C.-F. from the NSF under the grant DMS-0702871 is gratefully acknowledged. I.P. was partially supported by the NSF and the NSA.

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N2 - Based on the ideas in Ciocan-Fontanine, Konvalinka and Pak (2009) [5], we introduce the weighted analogue of the branching rule for the classical hook length formula, and give two proofs of this result. The first proof is completely bijective, and in a special case gives a new short combinatorial proof of the hook length formula. Our second proof is probabilistic, generalizing the (usual) hook walk proof of Greene, Nijenhuis and Wilf (1979) [15], as well as the q-walk of Kerov (1993) [20]. Further applications are also presented.

AB - Based on the ideas in Ciocan-Fontanine, Konvalinka and Pak (2009) [5], we introduce the weighted analogue of the branching rule for the classical hook length formula, and give two proofs of this result. The first proof is completely bijective, and in a special case gives a new short combinatorial proof of the hook length formula. Our second proof is probabilistic, generalizing the (usual) hook walk proof of Greene, Nijenhuis and Wilf (1979) [15], as well as the q-walk of Kerov (1993) [20]. Further applications are also presented.

KW - Bijective proof

KW - Hook-length formula

KW - Weighted analogue

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JO - Journal of Combinatorial Theory - Series A

JF - Journal of Combinatorial Theory - Series A

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ER -

The weighted hook length formula

Abstract

Bibliographical note

Keywords

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