Lattice paths and negatively indexed weight-dependent binomial coefficients

15.06.2021 15:15 - 16:45

Josef Küstner

Abstract: In 1992, Daniel Loeb considered a natural extension of the binomial coefficients to negative entries and gave a combinatorial interpretation in terms of hybrid sets. He showed that many of the fundamental properties of binomial coefficients continue to hold in this extended setting. Recently, Sam Formichella and Armin Straub showed that these results can be extended to the $q$-binomial coefficients with arbitrary integer values and extended the work of Loeb further by examining arithmetic properties of the $q$-binomial coefficients. In our work, we give an alternative combinatorial interpretation in terms of lattice paths and consider an extension of the more general weight-dependent binomial coefficients, first defined by Schlosser, to arbitrary integer values. Remarkably, many of the results of Loeb, Formichella and Straub continue to hold in the general weighted setting. In this talk I will also examine some important special cases of the weight-dependent binomial coefficients, including ordinary, $q$- and elliptic binomial coefficients as well as elementary and complete homogeneous symmetric functions.

This is joint work with Michael Schlosser and Meesue Yoo.

Meeting-ID: 945 4121 9182, Kenncode: Let2Vh


Ch. Krattenthaler

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