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.