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)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(16\ell <^>2,4\ell ;3)$$\end{document} Heffter space for any & ell;>= 1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell \ge 1$$\end{document}. Combining these constructions we obtain a shiftable (16 & ell;2mn,4 & ell;n;3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(16\ell <^>2mn,4\ell n;3)$$\end{document} Heffter space for every triple of positive integers (& ell;,m,n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\ell ,m,n)$$\end{document} with m >= n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m\ge n$$\end{document}.
Shiftable Heffter spaces
Buratti M.;Pasotti A.
2025-01-01
Abstract
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)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(16\ell <^>2,4\ell ;3)$$\end{document} Heffter space for any & ell;>= 1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell \ge 1$$\end{document}. Combining these constructions we obtain a shiftable (16 & ell;2mn,4 & ell;n;3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(16\ell <^>2mn,4\ell n;3)$$\end{document} Heffter space for every triple of positive integers (& ell;,m,n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\ell ,m,n)$$\end{document} with m >= n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m\ge n$$\end{document}.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
