Euler showed that the number of partitions of $n$ into distinct parts equals the number of partitions of $n$ into odd parts. This theorem was generalized by Glaisher and further by Franklin. Recently, Beck made three conjectures on partitions with restricted parts, which were confirmed analytically by Andrews and Chern and combinatorially by Yang. Analogous to Euler's partition theorem, it is known that the number of compositions of $n$ with odd parts equals the number of compositions of $n+1$ with parts greater than one, as both numbers equal the Fibonacci number $F_n$. Recently, Sills provided a bijective proof for this result using binary sequences, and Munagi proved a generalization similar to Glaisher's result using the zigzag graphs of compositions. Extending Sills' bijection, we obtain a further generalizaiton which is analogous to Franklin's result. We establish, both analytically and combinatorially, two closed formulas for the number of compositions with restricted parts appearing in our generalization. We also prove some composition analogues for the conjectures of Beck.

中文翻译：

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具有限制部分的组合物

欧拉证明了 $n$ 分割成不同部分的数量等于 $n$ 分割成奇数部分的数量。该定理由 Glaisher 和富兰克林进一步推广。最近，Beck 对有限制部分的分区做了三个猜想，得到了 Andrews 和 Chern 的分析证实以及 Yang 的组合证实。类似于欧拉的划分定理，已知$n$ 的奇数部分的组合数等于$n+1$ 的部分大于1 的组合数，因为这两个数字都等于斐波那契数$F_n$。最近，Sills 使用二元序列为这个结果提供了一个双射证明，而 Munagi 使用组合的锯齿形图证明了类似于 Glaisher 的结果的概括。扩展 Sills 的双射，我们得到了类似于富兰克林结果的进一步概括。我们在分析和组合上建立了两个封闭公式，用于在我们的概括中出现的具有限制部分的组合数。我们还证明了贝克猜想的一些组成类似物。