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Finite Kurepa tree is a finite rooted tree \({\mathbf {T}}_n= (T_n,\leq ,0)\) of height n in which each node of height \(k&lt;n_ splits="" into="" _k_1_="" successors.="" This="" tree="" of="" is="" related="" to="" the="" Kurepa="" left="" factorial="" hypothesis_="" which="" states="" that="" for="" no="" integer="" _n=""&gt;2\) , \(n\, |\, (0!+1!+2!+ \cdots + (n-1)!).\) We consider the hypothesis combinatorially studying partitions of \({\mathbf {T}}_{n-1}\) into n components having the same size. We proved that there are no such partitions if all of its components are connected subsets of \(T_{n-1}\) , or there is a component invariant under all automorphisms of \({\mathbf {T}}_{n-1}\) . We also found that \(\mathrm {Aut}({\mathbf {T}}_n)\) is the wreath product of factorial powers of permutation groups \({\mathbf {S}}_k\) , \(k\leq n\) .</n_>

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Partitions and Automorphisms of Finite Kurepa Trees

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  • 发表日期：

    2025年

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  • Žarko Mijajlović

摘要

Finite Kurepa tree is a finite rooted tree \({\mathbf {T}}_n= (T_n,\leq ,0)\) of height n in which each node of height \(k<n_ splits="" into="" _k_1_="" successors.="" This="" tree="" of="" is="" related="" to="" the="" Kurepa="" left="" factorial="" hypothesis_="" which="" states="" that="" for="" no="" integer="" _n="">2\) , \(n\, |\, (0!+1!+2!+ \cdots + (n-1)!).\) We consider the hypothesis combinatorially studying partitions of \({\mathbf {T}}_{n-1}\) into n components having the same size. We proved that there are no such partitions if all of its components are connected subsets of \(T_{n-1}\) , or there is a component invariant under all automorphisms of \({\mathbf {T}}_{n-1}\) . We also found that \(\mathrm {Aut}({\mathbf {T}}_n)\) is the wreath product of factorial powers of permutation groups \({\mathbf {S}}_k\) , \(k\leq n\) .