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 $i$, $1 \leq i \leq x$.