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Some Tilings, Colorings and Lattice Paths via Stern Polynomials

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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 use certain subsequences of two different but related types of generalized Stern polynomials to characterize all lattice paths, with specific restrictions, that go from the origin to the line \(x+y=n\) in the first quadrant of the xy-plane. The first kind of lattice paths can also be interpreted as tilings with squares and dominoes in one case and “black and white” colorings in another case. The second kind of lattice paths is certain weighted Delannoy paths; from our analysis, we obtain results on weighted Delannoy numbers and extensions with polynomial weights. Finally, we establish some connections with Jacobi polynomials.

Research supported in part by the Natural Sciences and Engineering Research Council of Canada, Grant # 145628481.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia, B3H 4R2, Canada

    Karl Dilcher

  2. P.O. Box 172, Millville, NJ, 08332-0172, USA

    Larry Ericksen

Authors
  1. Karl Dilcher
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  2. Larry Ericksen
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Correspondence to Larry Ericksen .

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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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Dilcher, K., Ericksen, L. (2019). Some Tilings, Colorings and Lattice Paths via Stern Polynomials. 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_11

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