Tǎrnǎuceanu and Bentea [M. Tǎrnǎuceanu, L. Bentea, On the number of fuzzy subgroups of finite abelian groups, Fuzzy Sets and Systems 159 (2008) 1084-1096] gave an explicit formula for the number of chains of subgroups in the lattice of a finite cyclic group by finding its generating function of one variable. Using this result Tǎrnǎuceanu [M. Tǎrnǎuceanu, Fuzzy subgroups of finite cyclic groups and Delannoy numbers, European J. Combin. 30 (2009) 283-287] found an explicit formula for the central Delannoy number. In this note we find a generating function of multi-variables for the number of chains of subgroups in the lattice of subgroups of a finite cyclic group. As results we give simplified formulas for the number of chains of subgroups in the lattice of subgroups of a finite cyclic group and for the central Delannoy numbers compared with the formulas given by Tǎrnǎuceanu and Bentea.
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机译：Tǎrnǎuceanu和Bentea [M. Tǎrnǎuceanu，L. Bentea，关于有限阿贝尔群的模糊子群的数量，Fuzzy Sets and Systems 159（2008）1084-1096]通过寻找，给出了一个有限循环群的格中子群链数的明确公式。它的一个变量的生成函数。使用该结果Tǎrnǎuceanu[M. Tǎrnǎuceanu，有限循环群和Delannoy数的Fuzzy子群，欧洲J. Combin。 30（2009）283-287]找到了中央Delannoy数的显式公式。在本说明中，我们发现了一个有限元群子群格中子群链数的多变量生成函数。结果，与Tǎrnǎuceanu和Bentea给出的公式相比，我们给出了有限循环组的子组格中的子组链数和中心Delannoy数的简化公式。
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