Constant Sum Partition of \(\{1,2,...,n\}\) Into Subsets With Prescribed Orders

Studies on partition of \(I_n\) = \(\{1, 2, . . . , n\}\) into subsets \(S_1, S_2, . . . , S_x\) so far considered with prescribed sum of the elements in each subset. In this paper, we study constant sum partitions \(\{S_1,S_2,...,S_x\}\) of \(I_n\) with prescribed \(|S_i|\), \(1 \leq i \leq x\). Theorem \ref{thm 2.3} is the main result which gives a necessary and sufficient condition for a partition set \(\{S_1,S_2,\ldots, S_x\}\) of \(I_n\) with prescribed \(|S_i|\) to be a constant sum partition of \(I_n\), \(1 \leq i \leq x\) and \(n > x \geq 2\). We state its applications in graph theory and also define {\em constant sum partition permutation} or {\em magic partition permutation} of \(I_n\). A partition \(\{S_1,S_2,\cdots,S_x\}\) of \(I_n\) is a {\em constant sum partition of \(I_n\)} if \(\sum_{j\in S_i}{j}\) is a constant for every \(i\), \(1 \leq i \leq x\).