Honam Mathematical Journal (호남수학학술지)

The Honam Mathematical Society (호남수학회)
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ENUMERATION OF FUSS-CATALAN PATHS BY TYPE AND BLOCKS

  • An, Suhyung (Department of Mathematics, Yonsei University) ;
  • Jung, JiYoon (Department of Mathematics, Marshall University) ;
  • Kim, Sangwook (Department of Mathematics, Chonnam National University)
  • Received : 2021.05.31
  • Accepted : 2021.09.08
  • Published : 2021.12.25
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Abstract

Armstrong enumerated the number of Fuss-Catalan paths with a given type and Rhoades provided the number of Dyck paths with a given type and a given number of blocks. In this paper we generalize those results to enumerate the number of Fuss-Catalan paths with a fixed type and a fixed number of blocks. We provide two proofs of this result. The first one uses the Chung-Feller theorem and a certain polynomial, while the second one is bijective. Also, we give a conjecture generalizing this result to the family of small Fuss-Schröder paths.

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Acknowledgement

Suhyung An was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education(NRF-2019R1I1A1A01059433)

References

  1. Suhyung An, JiYoon Jung, and Sangwook Kim. Enumeration of Fuss-Schroder paths. Electron. J. Combin., 24(2):Paper 2.30, 12, 2017.
  2. Drew Armstrong. Generalized noncrossing partitions and combinatorics of Coxeter groups. Mem. Amer. Math. Soc., 202(949):x+159, 2009.
  3. Sen-Peng Eu and Tung-Shan Fu. Lattice paths and generalized cluster complexes. J. Combin. Theory Ser. A, 115(7):1183-1210, 2008. https://doi.org/10.1016/j.jcta.2007.12.011
  4. G. Kreweras. Sur les partitions non croisees d'un cycle. Discrete Math., 1(4):333-350, 1972. https://doi.org/10.1016/0012-365X(72)90041-6
  5. George N. Raney. Functional composition patterns and power series reversion. Trans. Amer. Math. Soc., 94:441-451, 1960. https://doi.org/10.1090/S0002-9947-1960-0114765-9
  6. Brendon Rhoades. Enumeration of connected Catalan objects by type. European J. Combin., 32(2):330-338, 2011. https://doi.org/10.1016/j.ejc.2010.10.007
  7. Jiang Zeng. Multinomial convolution polynomials. Discrete Math., 160(1-3):219-228, 1996. https://doi.org/10.1016/0012-365X(95)00160-X