Distinguishing labelling of sets under the wreath product action of \(\overrightarrow{A_{n}}\)

In this article, it is proved that the distinguishing number for the action of \(\overrightarrow{A_{n}}\) $\overrightarrow{A_n}$ on the set \([2n]=\{1, 2,\ldots , 2n\}\) $[2n]=\{1,2,\dots ,2n\}$ is the \(n^{th}\) $n^{\mathit{th}}$ term of the sequence “1, 2, 3, 3, 4, 4, 4, 5,  ... (n appears \(n-1\) $n-1$ times prepended with 1)”. An optimal iterative algorithm to establish a closed formula to compute a distinguishing labelling of [2n] under the natural action of \(\overrightarrow{A_{n}}\) $\overrightarrow{A_n}$ is provided. Also, a closed formula to compute the distinguishing number for the above said action is established, in the sequel, it is observed that finding the distinguishing number for the action of \(\overrightarrow{A_{n}}\) $\overrightarrow{A_n}$ 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 exactly one box can contain only one pair of identical balls, but two pairs of identical balls can not be completely contained in two boxes with one pair in each box”.