偶数个最终常数映射与 nilpotent 对
Pairs of eventually constant maps and nilpotent pairs
AI总结：
本文研究了有限维向量空间间算子对的 nilpotent 性概率及最终常数映射的数量关系。
摘要：
Tom Leinster 给出了一种双射对应关系，将有限维向量空间 V 上的算子集与由 nilpotent 算子和 V 中向量组成的对集对应起来。在有限域上，这种双射暗示了算子为 nilpotent 的概率是 V 中向量数的倒数。我们把这个对应关系推广到两个向量空间之间的算子对，并确定了随机算子对为 nilpotent 的概率。我们还确定了这种构造的集合论对应关系，并计算了两个有限集合之间最终常数算子对的数量，这与完全二分图中的生成树数量密切相关。
英文摘要：
Tom Leinster gave a bijective correspondence between the set of operators on a finite-dimensional vector space $V$ and the set of pairs consisting of a nilpotent operator and a vector in $V$. Over a finite field this bijection implies that the probability that an operator be nilpotent is the reciprocal of the number of vectors in $V$. We generalize this correspondence to pairs of operators between pairs of vector spaces and determine the probability that a random pair of operators be nilpotent. We also determine the set-theoretical counterpart of this construction and compute the number of eventually constant pairs of maps between two finite sets, closely related to the number of spanning trees in a complete bipartite graph.
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。
