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A Note on a Theorem of Moser and Whitney
Published online by Cambridge University Press: 20 November 2018
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In a recent paper [1], L. Moser and E.L. Whitney have proved the following.
Theorem: The number of compositions of n into parts=1, 2, 4 or 5 (mod 6) and involving an even number of parts=4 or 5 (mod 6) exceeds by n the number of compositions of n into parts=1, 2, 4 or 5 (mod 6) and involving an odd number of parts=4 or 5 (mod 6).
Their method of proof utilizes the notion of weighted compositions and the method of generating series. They remark that they have not been able to find a direct combinatorial proof. The purpose of this note is to give a direct proof of a more general result.
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- Notes and Problems
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- Copyright © Canadian Mathematical Society 1962
References
1.
Moser, L. and Whitney, E.L., "Weighted Compositions", Canad. Math. Bull., Vol. 4(1961), pp. 39-43.10.4153/CMB-1961-006-0Google Scholar
2.
Miles, E.P. Jr, "Generalized Fibonacci Numbers and Associated Matrices", Amer. Math. Monthly, Vol. 67 (1960), pp. 745-752.10.1080/00029890.1960.11989593Google Scholar
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