On a divisor of the central binomial coefficient

Abstract

It is well known that for all $n\ge 1$ the number $n+1$ is a divisor of the central binomial coefficient ${2n\atopwithdelims ()n}$. Since the nth 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 ${2n\atopwithdelims ()n}$. We then show our main result follows from a more general observation regarding binomial coefficients ${n\atopwithdelims ()k}$ 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.