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Families of Parking Functions Counted by the Schröder and Baxter Numbers

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Lattice Path Combinatorics and Applications

Part of the book series: Developments in Mathematics ((DEVM,volume 58))

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Abstract

We define two new families of parking functions: one counted by Schröder numbers and the other by Baxter numbers. These families both include the well-known class of non-decreasing parking functions, which is counted by Catalan numbers and easily represented by Dyck paths, and they both are included in the class of underdiagonal sequences, which are bijective to permutations. We investigate their combinatorial properties exhibiting bijections between these two families and classes of lattice paths (Schröder paths and triples of non-intersecting lattice paths) and discovering a link between them and some classes of pattern avoiding permutations. Then, we provide a quite natural generalization for each of these families that results in some enumeration problems tackled by ECO method.

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Acknowledgements

The authors would like to thank Guillaume Chapuy for his valuable help on the study of the functional equation for \(GB_1\)-parking functions.

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Authors and Affiliations

  1. Labri, Université Bordeaux, Bordeaux, France

    Robert Cori

  2. LIAFA, Université Paris 7, Paris, France

    Enrica Duchi

  3. Dipartimento di Ingegneria dell’Informazione e Scienze Matematiche, University of Siena, Siena, Italy

    Veronica Guerrini & Simone Rinaldi

Authors
  1. Robert Cori
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  2. Enrica Duchi
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  3. Veronica Guerrini
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  4. Simone Rinaldi
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Corresponding author

Correspondence to Simone Rinaldi .

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Editors and Affiliations

  1. Department of Mathematics, The Pennsylvania State University, University Park, PA, USA

    George E. Andrews

  2. Fakultät für Mathematik, Universität Wien, Vienna, Austria

    Christian Krattenthaler

  3. Department of Mathematics and Statistics, California State Polytechnic University, Pomona, Pomona, CA, USA

    Alan Krinik

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Cori, R., Duchi, E., Guerrini, V., Rinaldi, S. (2019). Families of Parking Functions Counted by the Schröder and Baxter Numbers. In: Andrews, G., Krattenthaler, C., Krinik, A. (eds) Lattice Path Combinatorics and Applications. Developments in Mathematics, vol 58. Springer, Cham. https://doi.org/10.1007/978-3-030-11102-1_10

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