<p>The shiftable Heffter arrays are naturally generalized to the shiftable Heffter spaces. We present a recursive construction which, starting from a single shiftable Heffter space, leads to infinitely many other shiftable Heffter spaces of the same degree. We also present a direct construction making use of pandiagonal magic squares leading to a shiftable <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10623_2025_1657_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="81" /> </InlineMediaObject> <EquationSource Format="TEX">$(16\ell ^2,4\ell ;3)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mn>16</mn> <msup> <mi>ℓ</mi> <mn>2</mn> </msup> <mo>,</mo> <mn>4</mn> <mi>ℓ</mi> <mo>;</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> Heffter space for any <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10623_2025_1657_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">$\ell \ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ℓ</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. Combining these constructions we obtain a shiftable <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10623_2025_1657_Article_IEq3.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="116" /> </InlineMediaObject> <EquationSource Format="TEX">$(16\ell ^2mn,4\ell n;3)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mn>16</mn> <msup> <mi>ℓ</mi> <mn>2</mn> </msup> <mi>m</mi> <mi>n</mi> <mo>,</mo> <mn>4</mn> <mi>ℓ</mi> <mi>n</mi> <mo>;</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> Heffter space for every triple of positive integers <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10623_2025_1657_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="61" /> </InlineMediaObject> <EquationSource Format="TEX">$(\ell ,m,n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>ℓ</mi> <mo>,</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10623_2025_1657_Article_IEq5.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="51" /> </InlineMediaObject> <EquationSource Format="TEX">$m\ge n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>≥</mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation>.</p>

Shiftable Heffter spaces

期刊论文
发表日期：

2025年6月10日

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M. Buratti,
A. Pasotti

摘要

The shiftable Heffter arrays are naturally generalized to the shiftable Heffter spaces. We present a recursive construction which, starting from a single shiftable Heffter space, leads to infinitely many other shiftable Heffter spaces of the same degree. We also present a direct construction making use of pandiagonal magic squares leading to a shiftable $(16\ell ^2,4\ell ;3)$ $(16 ℓ^{2}, 4 ℓ; 3)$ Heffter space for any $\ell \ge 1$ $ℓ \geq 1$ . Combining these constructions we obtain a shiftable $(16\ell ^2mn,4\ell n;3)$ $(16 ℓ^{2} m n, 4 ℓ n; 3)$ Heffter space for every triple of positive integers $(\ell ,m,n)$ $(ℓ, m, n)$ with $m\ge n$ $m \geq n$ .