用相同的Betti曲线计数条形码
Counting Barcodes with the same Betti Curve
AI总结：
本文通过将持久同调与Kostant分区函数和魔法抛接序列的组合学联系起来，研究了如何通过Betti曲线计数不同的条形码。
摘要：
本文考虑了拓扑数据分析（TDA）中的一个重要逆问题：有多少种不同的条形码会产生相同的Betti曲线？等价地，给定一个函数β：[n]={1<⋯<n}→Z≥0，有多少种不同的方法可以将β表示为支持在[n]上的指示函数之和？我们回答这个问题的方法是将持久同调与Kostant分区函数的研究以及所谓的
英文摘要：
This paper considers an important inverse problem in topological data analysis (TDA): How many different barcodes produce the same Betti curve? Equivalently, given a function $β\colon [n]=\{1<\cdots< n\} \to \mathbb{Z}_{\geq 0}$, how many different ways can we write $β$ as a sum of indicator functions supported on intervals in $[n]$? Our answer to this question is to connect persistent homology with the study of the Kostant partition function and the enumerative combinatorics for so-called "magic" juggling sequences studied by Ronald Graham and others. Specifically, we prove an equivalence between our inverse problem and corresponding statements in these other two settings. From an applications and statistics point of view, our work provides a quantification of how lossy the TDA pipeline is when moving from persistent homology to persistent Betti numbers.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
