# Equsum Partitions of Sets of Positive Integers +
## 2 Initial Intervals of \(\mathbb{Z}^{+}\)
Finally, if \(\mathcal{B}\) and \(\mathcal{B}^{\prime}\) are both consesum, and \(\mathcal{B}^{\prime}\) is uniform, then \(\mathcal{B}^{\mathcal{R}}\oplus\mathcal{B}^{\prime}\) is equisum when \(\mathcal{B}^{\mathcal{R}}\), the _reverse_ of \(\mathcal{B}\), is defined by \(\mathcal{B}^{\mathcal{R}}:=\left\{B^{R}_{i}:=B_{b-i+1}\middle|1\leq i\leq b\right\}\). **Theorem 1**. : _Let \(b\in\mathbb{Z}^{+},\ b\geq 2.\) If \([1,\ n]\) has an equisum \(b\)-partition with defect \(\delta\), then so does \([1,\ n^{\prime}]\), for any \(n^{\prime}\in\mathbb{Z}^{+},\ n^{\prime}>n\), such that \(n^{\prime}\equiv n\pmod{2b}\)._  
Proof. : Let \(\mathcal{E}:=\{E_{i}|1\leq i\leq b\}\) be the trivial uniform \(b\)-partition of \([1,\,b]\) with \(E_{i}:=\{i\}\) for \(1\leq i\leq b\). Then \(\mathcal{E}\) is consesum, so \(\mathcal{A}:=\mathcal{E}\oplus\mathcal{E}^{\mathcal{R}}\) is a uniform equisum \(b\)-partition of \([1,\,2b]\). Put \(\mathcal{A}^{(1)}:=\mathcal{A}\) and \(\mathcal{A}^{(m+1)}:=\mathcal{A}\oplus\mathcal{A}^{(m)}\) for all \(m\in\mathbb{Z}^{+}.\) Then \(\mathcal{A}^{(m)}\) is a uniform equisum \(b\)-partition of \([1,\,2 mb]\). Let \(\mathcal{B}:=\{B_{i}|1\leq i\leq b\}\) be an equisum \(b\)-partition of \([1,\ n]\) with defect \(\delta\). Then, \(\mathcal{B}\oplus\mathcal{A}^{(m)}\) is an equisum \(b\)-partition of \([1,\,n^{\prime}:=\ n+2 mb]\) with defect \(\delta\). When \([1,\,n]\) has a \(b\)-partition \(\mathcal{B}\) of the specified type, the proof of Theorem 1 is, in effect, an unconditional algorithm for constructing from \(\mathcal{B}\) such a \(b\)-partition for each suitable \([1,\,n^{\prime}]\). The back and forth (boutsrophedon) construction producing \(\mathcal{A}^{(m)}\) can be described as _knitting_. With the convention that \(E_{1}-1:=\varnothing\), we _define_\(\mathcal{E}^{\prime}:=\mathcal{E}-1\) to be a consesum \(b\)-partition of \([1,b-1]\) with defect \(\delta=1\), admitting one empty block. The knitting step \(\mathcal{A}^{\prime}:=\mathcal{E}^{\prime}\oplus\mathcal{E}^{\mathcal{R}}\) produces an equisum \(b\)-partition of \([1,\,2b-1]\) with defect \(\delta=1\), so with ordinary knitting \(\mathcal{A}^{(m)}\) we have  
**Theorem 2**. : _For any \(b\in\mathbb{Z}^{+},\ b\geq 2\), and \(\delta\in\{0,\ 1\}\), the set \([1,n]\) has an equisum \(b\)-partition, with defect \(\delta\), for all \(n\in\mathbb{Z}^{+},\ n+\delta\equiv 0\pmod{2b}\)._  
A parallel construction, proceeding from consesum to equisum, now establishes. **Lemma 1**. : _Let \(b\in\mathbb{Z}^{+},\ b\geq 2.\) If \([1,\ n]\) has a consesum \(b\)-partition with defect \(\delta\), then \([1,\ n^{\prime}]\) has an equisum \(b\)-partition with defect \(\delta\) for any \(n^{\prime}\in\mathbb{Z}^{+},\ n^{\prime}>n\), such that \(n^{\prime}\equiv n+b\pmod{2b}\)._  
Proof. : As in the previous proof, \(\mathcal{E}\) is a uniform consesum \(b\)-partition of \([1,\ b]\), and \(\mathcal{A}:=\mathcal{E}\oplus\mathcal{E}^{\mathcal{R}}\) is a uniform equisum \(b\)-partition of \([1,\ 2b]\). For all \(m\in\mathbb{Z}^{+}\) let \(\mathcal{C}^{(1)}:=\mathcal{E},\mathcal{C}^{(m+1)}:=\mathcal{E}\oplus\mathcal{ A}^{(m)}.\) Then, \(\mathcal{C}^{(m)}\) is a uniform consesum \(b\)-partition of \([1,\ (2m-1)b]\). Suppose that \(\mathcal{B}:=\{B_{i}|1\leq i\leq b\}\) is a consesum \(b\)-partition of \([1,\ n]\) with defect \(\delta\). Then, \(\mathcal{B}^{\mathcal{R}}\oplus\mathcal{C}^{(m)}\) is an equisum \(b\)-partition of \([1,\ n+(2m-1)b]\) with defect \(\delta\). When \([1,\ n]\) has a consesum \(b\)-partition \(\mathcal{B}\) with defect \(\delta\), the proof of Lemma 1 is, in effect, an unconditional algorithm for constructing from \(\mathcal{B}\) an equisum \(b\)-partition with defect \(\delta\) for each suitable \([1,\ n^{\prime}]\). Knitting is the key. Let \(\mathcal{C}^{\prime(1)}:=\mathcal{E}^{\prime},\mathcal{C}^{\prime(m+1)}:= \mathcal{E}^{\prime}\oplus\mathcal{A}^{(m)}\), for all \(m\in\mathbb{Z}^{+}\).