Lattice paths and negatively indexed weight-dependent binomial coefficients

TL;DR

This paper extends the combinatorial and algebraic properties of binomial coefficients to negative and weighted cases using lattice paths, unifying various binomial variants like q- and elliptic binomials.

Contribution

It provides a new combinatorial interpretation for weight-dependent binomial coefficients at arbitrary integer values, generalizing prior results to a broader weighted setting.

Findings

Many properties of classical binomial coefficients hold in the weighted extension.

The framework unifies ordinary, q-, and elliptic binomial coefficients.

The results include combinatorial interpretations and algebraic identities.

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.