Weighted Lattice Paths Enumeration by Gaussian Polynomials

TL;DR

This paper explores the combinatorial interpretation of Gaussian polynomials via weighted lattice paths and board tilings, proving identities and evaluating sums to deepen understanding of their algebraic and combinatorial properties.

Contribution

It introduces new dual identities and sum evaluations for Gaussian polynomials using combinatorial models of weighted lattice paths and tilings.

Findings

Established dual identities for Gaussian polynomials

Evaluated new sums related to weighted lattice paths

Linked combinatorial models to algebraic properties

Abstract

The Gaussian polynomial in variable $q$ is defined as the $q$-analog of the binomial coefficient. In addition to remarkable implications of these polynomials to abstract algebra, matrix theory and quantum computing, there is also a combinatorial interpretation through weighted lattice paths. This interpretation is equivalent to weighted board tilings, which can be used to establish Gaussian polynomial identities. In particular, we prove duals of such identities and evaluate related sums.