论文标题
单位高管的分解和广义莫比乌斯系列的恢复
Decompositions of Unit Hypercubes and the Reversion of a Generalized Möbius Series
论文作者
论文摘要
令$ s_d（n）$为$ d $ d $二维超立方体的不同分解数量，该分解可以通过$ n $矩形区域，可以通过一系列分裂操作获得。我们证明，生成系列$ y = \ sum_ {n \ geq 1} s_d（n）x^n $满足功能方程$ x = \ sum_ {n \ geq 1}μ_d（n）y^n $，其中$μ_d（n）$是$ d $ -d $ -d $ -fold dirichlet dirichlet dirichlet the the the the the the the the the dirichlet dirichlet vermuip of the the the the the the the n of。这概括了Goulden等人的最新结果，并表明$ s_1（n）$还提供了$ \ mz $的天然精确覆盖系统的数量，并提供$ n $剩余类。我们还证明了$ s_d（n）$的渐近公式，并描述了$ 1 $维分解和天然精确覆盖系统之间的两次射击。
Let $s_d(n)$ be the number of distinct decompositions of the $d$-dimensional hypercube with $n$ rectangular regions that can be obtained via a sequence of splitting operations. We prove that the generating series $y = \sum_{n \geq 1} s_d(n)x^n$ satisfies the functional equation $x = \sum_{n\geq 1} μ_d(n)y^n$, where $μ_d(n)$ is the $d$-fold Dirichlet convolution of the Möbius function. This generalizes a recent result by Goulden et al., and shows that $s_1(n)$ also gives the number of natural exact covering systems of $\mZ$ with $n$ residual classes. We also prove an asymptotic formula for $s_d(n)$ and describe a bijection between $1$-dimensional decompositions and natural exact covering systems.
