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An algebra over the operad of posets and structural binomial identities

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Abstract

We study generating functions of strict and non-strict order polynomials of series–parallel posets, called order series. These order series are closely related to Ehrhart series and \(h^*\)-polynomials of the associated order polytopes. We explain how they can be understood as algebras over a certain operad of posets. Our main results are based on the fact that the order series of chains form a basis in the space of order series. This allows to reduce the search space of an algorithm that finds for a given power series f(x), if possible, a poset P such that f(x) is the generating function of the order polynomial of P. In terms of Ehrhart theory of order polytopes, the coordinates with respect to this basis describe the number of (internal) simplices in the canonical triangulation of the order polytope of P. Furthermore, we derive a new proof of the reciprocity theorem of Stanley. As an application, we find new identities for binomial coefficients and for finite partitions that allow for empty sets, and we describe properties of the negative hypergeometric distribution.

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Acknowledgements

We thank Elton P. Hsu for carefully reading the section on probability. We thank Max Hopkings, Isaias Marin Gaviria, and Mario Sanchez for several discussions. We thank John Baez for online discussions about binomial identities and Theo Johnson-Freyd for discussions about the units of the operad of series–parallel posets. Furthermore, we thank Raman Sanyal for suggesting the reference [4], and the anonymous referee(s) for many helpful comments and for pointing us to the papers [11, 24, 30].

Funding

The second author was funded through the Royal Society grant URF\(\setminus\)R1\(\setminus\)20147. The third author was funded by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2020R1C1C1A0100826). This project started while the third author worked at NewSci Labs and he thanks them for their hospitality.

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Authors and Affiliations

  1. División de Ingenierías, Universidad de Guanajuato, Guanajuato, Mexico

    José Antonio Arciniega-Nevárez

  2. Mathematical Institute, University of Oxford, Oxford, UK

    Marko Berghoff

  3. Department of Mathematics, Yonsei University, Seoul, Korea

    Eric Rubiel Dolores-Cuenca

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  1. José Antonio Arciniega-Nevárez
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  2. Marko Berghoff
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  3. Eric Rubiel Dolores-Cuenca
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Correspondence to José Antonio Arciniega-Nevárez.

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Arciniega-Nevárez, J.A., Berghoff, M. & Dolores-Cuenca, E.R. An algebra over the operad of posets and structural binomial identities. Bol. Soc. Mat. Mex. 29, 8 (2023). https://doi.org/10.1007/s40590-022-00478-9

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