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Bivariate Order Polynomials

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Abstract

Motivated by Dohmen–Pönitz–Tittmann’s bivariate chromatic polynomial \(\chi _G(x,y)\), which counts all x-colorings of a graph G such that adjacent vertices get different colors if they are \(\le y\), we introduce a bivarate version of Stanley’s order polynomial, which counts order preserving maps from a given poset to a chain. Our results include decomposition formulas in terms of linear extensions, a combinatorial reciprocity theorem, and connections to bivariate chromatic polynomials.

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Notes

  1. A reciprocity theorem for the bivariate chromatic polynomial was previously given in [2]; unfortunately, its statement and proof are wrong.

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Authors and Affiliations

  1. Department of Mathematics, San Francisco State University, San Francisco, CA, 94132, USA

    Matthias Beck

  2. Department of Mathematics, Mills College, Oakland, CA, 94613, USA

    Maryam Farahmand

  3. College of Alameda, 555 Ralph Appezzato Memorial Parkway, Alameda, CA, 94501, USA

    Gina Karunaratne

  4. Graduate School of Education, University of California-Berkeley, Berkeley, CA, 94720, USA

    Sandra Zuniga Ruiz

Authors
  1. Matthias Beck
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  2. Maryam Farahmand
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  3. Gina Karunaratne
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  4. Sandra Zuniga Ruiz
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Correspondence to Maryam Farahmand.

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Beck, M., Farahmand, M., Karunaratne, G. et al. Bivariate Order Polynomials. Graphs and Combinatorics 36, 921–931 (2020). https://doi.org/10.1007/s00373-019-02128-w

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  • DOI: https://doi.org/10.1007/s00373-019-02128-w

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