On distinguishing labelling of sets under the wreath product action

Madhu Dadhwal () and Pankaj ()
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Madhu Dadhwal: Himachal Pradesh University
Pankaj: Himachal Pradesh University

Indian Journal of Pure and Applied Mathematics, 2023, vol. 54, issue 3, 920-935

Abstract: Abstract In this paper, it is proved that the distinguishing number for the action of $$\overrightarrow{S_{n}}$$ S n → on the set $$[2n]=\{1, 2, \ldots , 2n\}$$ [ 2 n ] = { 1 , 2 , … , 2 n } is one more than the nth term of the sequence “ $$1, 2, 2, 3, 3, 3, 4,\ldots $$ 1 , 2 , 2 , 3 , 3 , 3 , 4 , … (n appears n times)”. Further, the cycle decomposition of the elements of $$\overrightarrow{S_{n}}$$ S n → is completely characterized. Also, the correspondence between the distinguishing labelling for the action of a group G on a G-set X and a partition of g-invariant subsets of X for every $$g\in G\setminus Stab_{G}(X)$$ g ∈ G \ S t a b G ( X ) , is established. In the sequel, we obtain the minimum possible elements of $$\overrightarrow{S_{n}}$$ S n → affecting the invariance of the partitioning components in a partition of the set X under the action of non identity elements of $$\overrightarrow{S_{n}}$$ S n → and as a consequence of this, it is observed that finding the distinguishing number for the action of $$\overrightarrow{S_{n}}$$ S n → on the set [2n] is equivalent to answering a combinatorial problem which states that “what is the minimum number of boxes required to arrange n distinct pair of identical balls in such a way that neither a pair of identical balls is contained in the same box nor any two boxes contain two pairs of identical balls completely”. We also provide an optimal algorithm to establish a formula to compute a distinguishing labelling of [2n] under the natural action of $$\overrightarrow{S_{n}}$$ S n → .

Keywords: Cycle decomposition; Distinguishing number; Distinguishing group actions; Distinguishing labelling of sets; 20D60; 05C25; 20B35 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s13226-022-00314-w

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