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- W3176556317 abstract "It is well known that for all $$nge 1$$ the number $$n+1$$ is a divisor of the central binomial coefficient $${2natopwithdelims ()n}$$ . 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 $${2natopwithdelims ()n}$$ . We then show our main result follows from a more general observation regarding binomial coefficients $${natopwithdelims ()k}$$ 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." @default.
- W3176556317 created "2021-07-05" @default.
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- W3176556317 date "2021-06-28" @default.
- W3176556317 modified "2023-09-27" @default.
- W3176556317 title "On a divisor of the central binomial coefficient" @default.
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- W3176556317 doi "https://doi.org/10.1007/s11139-021-00429-4" @default.
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