2021/02/01 by Just, Matthew, Schneider, Maxwell · 1 citation
#05A10 #11B65 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2102.00944
It is well known that for all n≥1 the number n+ 1 is a divisor of the central binomial coefficient 2n\choose 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\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.