On alternating sums of binomial coefficients and $q$-binomial coefficients

TL;DR

This paper evaluates alternating sums of binomial coefficients using combinatorial methods and extends these results to $q$-analogues involving $q$-binomial coefficients, combining combinatorics and partition theory.

Contribution

It introduces new combinatorial proofs for alternating binomial sums and derives novel $q$-analogues using partition theoretic techniques.

Findings

Explicit formulas for alternating sums of binomial coefficients

New $q$-analogues involving $q$-binomial coefficients

Integration of combinatorial and partition methods

Abstract

In this paper we shall evaluate two alternating sums of binomial coefficients by a combinatorial argument. Moreover, by combining the same combinatorial idea with partition theoretic techniques, we provide $q$-analogues involving the $q$-binomial coefficients.