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A Note on a Theorem of Moser and Whitney

Published online by Cambridge University Press:  20 November 2018

T. V. Narayana  and
H. M. Pettigrew
T. V. Narayana
Affiliation:
National Institutes of Health
H. M. Pettigrew
Affiliation:
National Institutes of Health
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Extract

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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.

Type
Notes and Problems
Information
Canadian Mathematical Bulletin , Volume 5 , Issue 2 , May 1962 , pp. 191 - 194
Copyright
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