Partitions and Automorphisms of Finite Kurepa Trees

Žarko Mijajlović ()
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Žarko Mijajlović: Faculty of Mathematics

Chapter Chapter 12 in Analysis, Approximation, Optimization: Computation and Applications, 2025, pp 211-234 from Springer

Abstract: Abstract Finite Kurepa tree is a finite rooted tree T n = ( T n , ≤ , 0 ) $${\mathbf {T}}_n= (T_n,\leq ,0)$$ of height n in which each node of height k 2 $$n>2$$ , n | ( 0 ! + 1 ! + 2 ! + ⋯ + ( n −1 ) ! ) . $$n\, |\, (0!+1!+2!+ \cdots + (n-1)!).$$ We consider the hypothesis combinatorially studying partitions of T n −1 $${\mathbf {T}}_{n-1}$$ 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 $$T_{n-1}$$ , or there is a component invariant under all automorphisms of T n −1 $${\mathbf {T}}_{n-1}$$ . We also found that Aut ( T n ) $$\mathrm {Aut}({\mathbf {T}}_n)$$ is the wreath product of factorial powers of permutation groups S k $${\mathbf {S}}_k$$ , k ≤ n $$k\leq n$$ .

Date: 2025
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DOI: 10.1007/978-3-031-85743-0_12

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