广义Hultman数与断点图的圈结构
Generalized Hultman Numbers and Cycle Structures of Breakpoint Graphs
摘要：
基因组重排可以被建模为$k$ -断裂，其在k个位置处断裂基因组并将所得片段以新的顺序粘合。特别地，反转、易位、融合和分裂被建模为$2$ -断裂，并且易位被建模为$3$ -断裂。虽然在进化中尚未观察到$k>3$的$k$ -断裂重排，但它们在癌症基因组学中用于模拟染色体断裂，这是一种基因组中同时发生多个断裂的灾难性事件。已知两个基因组之间的$k$ -断裂距离（即，将一个基因组转化为另一个基因组所需的最小$k$ -断裂数）可以根据这些基因组的断裂点图中的循环长度来计算。在目前的工作中，我们解决的组合问题，在一个给定的$k$中断距离从一个固定的unichromosomal基因组的基因组计数。更一般地，我们列举基因组对，其断点图具有给定的周期长度分布。我们进一步展示了我们的枚举如何可以用于随机基因组的均匀采样在一个给定的$k$ -中断距离，并描述其连接到各种组合对象，如贝尔多项式。
英文摘要：
Genome rearrangements can be modeled as $k$-breaks, which break a genome at k positions and glue the resulting fragments in a new order. In particular, reversals, translocations, fusions, and fissions are modeled as $2$-breaks, and transpositions are modeled as $3$-breaks. While $k$-break rearrangements for $k>3$ have not been observed in evolution, they are used in cancer genomics to model chromothripsis, a catastrophic event of multiple breakages happening simultaneously in a genome. It is known that the $k$-break distance between two genomes (i.e., the minimum number of $k$-breaks required to transform one genome into the other) can be computed in terms of cycle lengths in the breakpoint graph of these genomes. In the current work, we address the combinatorial problem of enumerating genomes at a given $k$-break distance from a fixed unichromosomal genome. More generally, we enumerate genome pairs, whose breakpoint graph has a given distribution of cycle lengths. We further show how our enumeration can be used for uniform sampling of random genomes at a given $k$-break distance, and describe its connection to various combinatorial objects such as Bell polynomials.
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