On distinguishing labelling of sets under the wreath product action

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 → .