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Vacillating Hecke tableaux and linked partitions

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Abstract

We introduce the structure of vacillating Hecke tableaux. By using the Hecke insertion algorithm developed by Buch, Kresch, Shimozono, Tamvakis and Yong, we establish a one-to-one correspondence between vacillating Hecke tableaux and linked partitions, which arise in free probability theory. We define a Hecke diagram as a Young diagram possibly with a marked corner. A vacillating Hecke tableau is defined as a sequence of Hecke diagrams subject to a certain condition on addition and deletion of rook strips. The notion of a rook strip was introduced by Buch in the study of the Littlewood–Richardson rule for stable Grothendieck polynomials. We show that the crossing number and the nesting number of a linked partition can be determined by the maximal number of rows and the maximal number of columns of diagrams in the corresponding vacillating Hecke tableau. The proof relies on a theorem due to Thomas and Yong. As consequences, we confirm two conjectures on the distribution of the crossing number and the nesting number over linked partitions and ordinary partitions, respectively proposed by de Mier and Kim.

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Acknowledgments

We wish to thank the referee for valuable suggestions. This work was supported by the 973 Project, the PCSIRT Project of the Ministry of Education, the National Science Foundation of China, the Science Foundation of Hebei Province and Grant No. BR2-231 of the Education Commission of Hebei Province.

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Authors and Affiliations

  1. Center for Applied Mathematics, Tianjin University, Tianjin, 300072, People’s Republic of China

    William Y. C. Chen

  2. Center for Combinatorics, LPMC-TJKLC, Nankai University, Tianjin, 300071, People’s Republic of China

    Peter L. Guo

  3. College of Mathematics and Statistics, Hebei University of Economics and Business, Shijiazhuang, Hebei, 050061, People’s Republic of China

    Sabrina X. M. Pang

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  1. William Y. C. Chen
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  2. Peter L. Guo
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  3. Sabrina X. M. Pang
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Correspondence to Peter L. Guo.

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Chen, W.Y.C., Guo, P.L. & Pang, S.X.M. Vacillating Hecke tableaux and linked partitions. Math. Z. 281, 661–672 (2015). https://doi.org/10.1007/s00209-015-1501-0

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  • DOI: https://doi.org/10.1007/s00209-015-1501-0

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