On a divisor of the central binomial coefficient

TL;DR

This paper investigates divisibility properties of central binomial coefficients, extends combinatorial partition results inspired by the Chung-Feller theorem, and offers a new interpretation of Catalan numbers.

Contribution

It introduces a novel combinatorial partition related to the divisor 2n-1 of the central binomial coefficient and generalizes results for binomial coefficients with coprime parameters.

Findings

Partitioning of lattice paths for divisor 2n-1 is achieved.

Main results derived from properties of binomial coefficients with coprime n and k.

Discussion includes limitations when n and k are not coprime.

Abstract

It is well known that for all $n\geq1$ the number $n+ 1$ is a divisor of the central binomial coefficient ${2n\choose n}$. Since the $n$th central binomial coefficient equals the number of lattice paths from $(0,0)$ to $(n,n)$ by unit steps north or east, a natural question is whether there is a way to partition these paths into sets of $n+ 1$ paths or $n+1$ equinumerous sets of paths. The Chung-Feller theorem gives an elegant answer to this question. We pose and deliver an answer to the analogous question for $2n-1$, another divisor of ${2n\choose n}$. We then show our main result follows from a more general observation regarding binomial coefficients ${n\choose k}$ with $n$ and $k$ relatively prime. A discussion of the case where $n$ and $k$ are not relatively prime is also given, highlighting the limitations of our methods. Finally, we come full circle and give a novel interpretation of the Catalan numbers.