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DOI: 10.4230/LIPIcs.SoCG.2025.16

Signotopes with Few Plus Signs

Authors: Helena Bergold, Lukas Egeling, and Hung P. Hoang

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract

Arrangements of pseudohyperplanes are widely studied in computational geometry. A rich subclass of pseudohyerplane arrangements, which has gained more attention in recent years, is the so-called signotopes. Introduced by Manin and Schechtman (1989), the higher Bruhat order is a natural order of r-signotopes on n elements, with the signotope corresponding to the cyclic arrangement as the minimal element. In this paper, we show that the lower (and by symmetry upper) levels of this higher Bruhat order contain the same number of elements for a fixed difference n-r. This result implies that given the difference d = n-r and p, the number of one-element extensions of the cyclic arrangement of n hyperplanes in ℝ^d with at most p points on one side of the extending pseudohyperplane does not depend on n, as long as n ≥ d + p.

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Helena Bergold, Lukas Egeling, and Hung P. Hoang. Signotopes with Few Plus Signs. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 16:1-16:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bergold_et_al:LIPIcs.SoCG.2025.16,
  author =	{Bergold, Helena and Egeling, Lukas and Hoang, Hung P.},
  title =	{{Signotopes with Few Plus Signs}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{16:1--16:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.16},
  URN =		{urn:nbn:de:0030-drops-231681},
  doi =		{10.4230/LIPIcs.SoCG.2025.16},
  annote =	{Keywords: flip graph, higher Bruhat order, signotope, counting, Ferrers diagram, one-element extension}
}

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Signotopes with Few Plus Signs

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