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.

众所周知，对于所有\(n\ge 1\)数\(n+1\)是中心二项式系数\({2n\atopwithdelims ()n}\)的除数。由于第n个中心二项式系数等于从 (0, 0) 到 ( n ,  n ) 向北或向东单位步长的晶格路径数，一个自然的问题是是否有办法将这些路径划分为\( n+1\)条路径或\(n+1\)组路径。Chung-Feller 定理给出了这个问题的优雅答案。我们为\(2n-1\)的类似问题提出并提供了答案，这是\({2n\atopwithdelims ()n}\) 的另一个除数. 然后，我们展示了我们的主要结果来自关于二项式系数\({n\atopwithdelims ()k}\)的更一般的观察，其中n和k互为素数。还讨论了n和k不是互质数的情况，突出了我们方法的局限性。最后，我们又回到原点，对加泰罗尼亚数字进行了新颖的解释。