Scholarlink

<p>Tom Leinster gave a bijective correspondence between the set of operators on a finite-dimensional vector space <i>V</i> and the set of pairs consisting of a nilpotent operator and a vector in <i>V</i>. Over a finite field this bijection implies that the probability that an operator be nilpotent is the reciprocal of the number of vectors in <i>V</i>. We generalize this correspondence to pairs of operators in opposite directions 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.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

Pairs of eventually constant maps and nilpotent pairs

  • 期刊论文

  • 发表日期：

    2026年7月4日

查看全文
  • Weixi Chen,
  • Mee Seong Im,
  • Mikhail Khovanov,
  • Catherine Lillja,
  • Nicolas Rugo

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 in opposite directions 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.