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

Abstract

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 $|S_i|$, $1\le i\le x$. Theorem \ref{thm 2.3} is the main result which gives a necessary and sufficient condition for a partition set $\{S_1,S_2,\dots ,S_x\}$ of $I_n$ with prescribed $|S_i|$ to be a constant sum partition of $I_n$, $1\le i\le x$ and $n>x\ge 2$. 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 $i$, $1\le i\le x$.

13 pages