# Equsum Partitions of Sets of Positive Integers +
## 2 Initial Intervals of \(\mathbb{Z}^{+}\)  
The most natural class of set \(V\) to consider for equisum partition is an initial interval of integers  
\[[1,\ n]:=\big{\{}a\in\mathbb{Z}^{+}:a\leq n\big{\}}.\]  
If \(\mathcal{B}:=\{B_{i}|1\leq i\leq b\}\) is a \(b\)-partition of \(V:=\ [1,\ n]\), its range of block sizes \(|B_{i}|:=k_{i},\ 1\leq i\leq b\) is of interest. The _defect_ of \(\mathcal{B}\) is the smallest integer \(\delta\) such that  
\[\big{|}k_{i}-k_{j}\big{|}\leq\delta\iff i\
eq j.\]  
If \(\mathcal{B}\) has defect \(\delta=0\) it is a _uniform_ partition of \(V\), with _block size_\(k:=n/b\). It is natural to seek equisum partitions with minimum defect. Two examples serve to give insight into equisum partitions of \([1,\ n]\). **Example 1**. : _If \(V=[1,\ 14]\) then \(\Sigma V=105\), so any proper equisum partition of \(V\) must have order \(b\in\{3,\ 5,\ 7\}\). Since \(b=7\) is the only possible order that is a factor of \(14\), it is the only case where defect \(\delta=0\) is possible. An equisum \(b\)-partition does exist in each case:_  
\[ \begin{array}{l}b=7,\ s=15:\ \{1,\ 14\},\{2,\ 13\},\{3,\ 12\},\{4,\ 11\},\{5,\ 10\},\{6,\ 9\},\{7,\ 8\};\\ b=5,\ s=21:\ \{7,\ 14\},\{1,\ 8,\ 12\},\{2,\ 9,\ 10\},\{3,\ 5,\ 13\},\{4,\ 6,\ 11\};\\ b=3,\ s=35:\ \{4,\ 8,\ 10,\ 13\},\{1,\ 5,\ 6,\ 9,\ 14\},\{2,\ 3,\ 7,\ 11,\ 12\}.\end{array} \]  
_The first partition is uniform, the other two have defect \(\delta=1\)._  
**Example 2**. : _If \(V=[1,\ 15]\) then \(\Sigma V=120\), so any proper equisum partition of \(V\) has order \(b\in\{2,\ 3,\ 4,\ 5,\ 6,\ 8\}\). An equisum \(b\)-partition exists in each case. For instance,_  
\[b=8,\ s=15:\ \{15\},\{1,\ 14\},\{2,\ 13\},\{3,\ 12\},\{4,\ 11\},\{5,\ 10\},\{6,\ 9\},\{7,\ 8\}.\]  
_By taking unions of consecutive pairs of blocks, then repeating, equisum partitions of orders \(4\) and \(2\) are immediately produced. All three have defect \(\delta=1\). This is best possible, since each has even order but \(n=15\) is odd._  
_Next, for the two possible orders which are divisors of \(15\) there are uniform equisum partitions:_  
\[ \begin{array}{l}b=5,\ s=24:\ \{1,\ 8,\ 15\},\{2,\ 9,\ 13\},\{3,\ 10,\ 11\},\{4,\ 6,\ 14\},\{5,\ 7,\ 12\};\\ b=3,\ s=40:\ \{1,\ 5,\ 9,\ 11,\ 14\},\{2,\ 6,\ 7,\ 10,\ 15\},\{3,\ 4,\ 8,\ 12,\ 13\}.\end{array} \]  
_Finally, for order \(6\) we have an equisum partition with defect \(\delta=1\):_  
\[b=6,\ s=20:\ \{5,\ 15\},\{6,\ 14\},\{8,\ 12\},\{1,\ 9,\ 10\},\{2,\ 7,\ 11\},\{3,\ 4,\ 13\}.\]  
_Notice that pairs of blocks of different sizes in this equisum \(6\)-partition could be combined to give an alternative uniform equisum \(3\)-partition._  
We now introduce a construction which serves to generalise these examples. For any \(n\in\mathbb{Z}^{+}\) and \(X\subseteq\mathbb{Z}^{+}\), define \(X+n:=\{x+n|x\in X\}\). **Direct Sum Construction:** _Let \(\mathcal{B}:=\{B_{i}|1\leq i\leq b\}\) and \(\mathcal{B}^{\prime}:=\left\{B^{\prime}_{i}|1\leq i\leq b\right\}\) be \(b\)-partitions of the sets \(V:=[1,\ n]\) and \(V^{\prime}:=[1,\ n^{\prime}]\), respectively. The direct sum_  
\[\mathcal{B}^{\prime}:=\mathcal{B}\oplus\mathcal{B}^{\prime}:=\left\{B_{i}\cup \left(B^{\prime}_{i}+n\right)\middle|1\leq i\leq b\right\}\]  
_is a \(b\)-partition of the set \(V^{\prime\prime}:=[1,\ n+n^{\prime}]\)._The direct sum construction has several useful special cases. If \(\mathcal{B}\) and \(\mathcal{B}^{\prime}\) are uniform, then \(\mathcal{B}^{\prime}\) is uniform. If \(\mathcal{B}\) and \(\mathcal{B}^{\prime}\) are equisum, and \(\mathcal{B}^{\prime}\) is uniform, then \(\mathcal{B}^{\prime\prime}\) is equisum. Let us call \(\mathcal{B}:=\{B_{i}|1\leq i\leq b\}\) a _consecsum_\(b\)-partition of \(V:=[1,\,n]\) if its block sums \(\Sigma B_{i}:=s_{i},\,1\leq i\leq b\) satisfy \(s_{i+1}=s_{i}+1\) for \(1\leq i<b\). We now note two more special cases of the direct sum construction. If \(\mathcal{B}\) is consesum and \(\mathcal{B}^{\prime}\) is uniform equisum, then \(\mathcal{B}^{\prime\prime}\) is consesum.