The distribution of the Carlitz binomial coefficients modulo a prime

TL;DR

This paper develops a method to count how Carlitz binomial coefficients distribute among residue classes modulo a prime in function fields, extending classical combinatorial theorems to algebraic structures over finite fields.

Contribution

It introduces a novel approach for analyzing the distribution of Carlitz binomial coefficients modulo primes in function fields, serving as a function field analogue of Garfield-Wilf theorem.

Findings

Provides a formula for counting coefficients in each residue class

Establishes a function field analogue of Garfield-Wilf theorem

Enhances understanding of binomial coefficient distributions in algebraic structures

Abstract

For a nonnegative integer $n$, and a prime $\wp$ in $\mathbb{F}_q[T]$, we prove a result that provides a method for computing the number of integers $m$ with $0 \le m \le n$ for which the Carlitz binomial coefficients $\binom{n}{m}_C$ fall into each of the residue classes modulo $\wp$. Our main result can be viewed as a function field analogue of the Garfield-Wilf theorem.