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Polynomization of the Bessenrodt–Ono Type Inequalities for A-Partition Functions

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Abstract

For an arbitrary set or multiset A of positive integers, we associate the A-partition function \(p_A(n)\) (that is the number of partitions of n whose parts belong to A). We also consider the analogue of the k-colored partition function, namely, \(p_{A,-k}(n)\). Further, we define a family of polynomials \(f_{A,n}(x)\) which satisfy the equality \(f_{A,n}(k)=p_{A,-k}(n)\) for all \(n\in \mathbb {Z}_{\ge 0}\) and \(k\in \mathbb {N}\). This paper concerns a polynomialization of the Bessenrodt–Ono inequality, namely

$$\begin{aligned} f_{A,a}(x)f_{A,b}(x)>f_{A,a+b}(x), \end{aligned}$$

where a, b are positive integers. We determine efficient criteria for the solutions of this inequality. Moreover, we also investigate a few basic properties related to both functions \(f_{A,n}(x)\) and \(f_{A,n}'(x)\).

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Acknowledgements

The first author would like to thank Piotr Miska, Maciej Ulas, and Błażej Żmija for their valuable comments and helpful suggestions. His research was funded by a grant of the National Science Centre (NCN), Poland, no. UMO-2019/34/E/ST1/00094. Finally, we thank the anonymous reviewers for their careful reading of our manuscript and their many insightful comments and suggestions.

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Author notes

  1. Bernhard Heim and Markus Neuhauser contributed equally to this work.

Authors and Affiliations

  1. Institute of Mathematics, Faculty of Mathematics and Computer Science, Jagiellonian University, S. Łojasiewicza 6, 30-348, Kraków, Poland

    Krystian Gajdzica

  2. Mathematical Institute, Faculty of Mathematical and Natural Sciences, University of Cologne, Weyertal 86–90, 50931, Cologne, Germany

    Bernhard Heim

  3. Kutaisi International University, Youth Avenue 5/7, 4600, Kutaisi, Georgia

    Markus Neuhauser

  4. Lehrstuhl A für Mathematik, RWTH Aachen University, 52056, Aachen, Germany

    Bernhard Heim & Markus Neuhauser

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  1. Krystian Gajdzica
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Correspondence to Krystian Gajdzica.

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Communicated by Ken Ono

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Gajdzica, K., Heim, B. & Neuhauser, M. Polynomization of the Bessenrodt–Ono Type Inequalities for A-Partition Functions. Ann. Comb. (2024). https://doi.org/10.1007/s00026-024-00692-4

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