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Combinatorial proof of Deddens’ theorem. (Russian) Zbl 0615.10004
Algebraic systems with one operation and one relation, Interuniv. Collect. sci. Works, Leningrad 1985, 46-53 (1985).
[For the entire collection see Zbl 0592.00016.]
Let m and n be relatively prime positive integers. Put $$G=\{um+vn$$; u and v are arbitrary non-negative integers$$\}$$. For any $$g\in G$$ we denote by N(j,g) the number of ways of representing g as the sum of j non-zero elements of G. Put $$L(0)=1$$ and $$L(g)=\sum^{\infty}_{j=1}(-1)^ j N(j,g)$$ for $$g\in G$$, $$g\neq 0.$$
The author gives a combinatorial proof of the following theorem of J. A. Deddens [J. Comb. Theory, Ser. A 26, 189-192 (1979; Zbl 0414.05005)]: If $$g\equiv 0$$ or $$m+n$$ (mod mn), then $$L(g)=1$$. If $$g\equiv m$$ or n (mod mn), then $$L(g)=-1$$. Otherwise $$L(g)=0$$.
Reviewer: B.Pondeliček

MSC:
 11A07 Congruences; primitive roots; residue systems 11A25 Arithmetic functions; related numbers; inversion formulas 06F05 Ordered semigroups and monoids