Lattice paths and negatively indexed weight-dependent binomial coefficients
(36 pages)
Abstract.
In 1992, 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, Formichella
and 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 this paper, 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 the second
author, to arbitrary integer values. Remarkably, many of the results of Loeb,
Formichella and Straub continue to hold in the general weighted setting. We
also examine 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.
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