Arrangements of pseudolines are classic objects in discrete and computational geometry. They have been studied with increasing intensity since their introduction almost 100 years ago. The study of the number B_n of non-isomorphic simple arrangements of n pseudolines goes back to Goodman and Pollack, Knuth, and others. It is known that B_n is in the order of 2^Θ(n²) and finding asymptotic bounds on b_n = log₂(B_n)/n² remains a challenging task. In 2011, Felsner and Valtr showed that 0.1887 ≤ b_n ≤ 0.6571 for sufficiently large n. The upper bound remains untouched but in 2020 Dumitrescu and Mandal improved the lower bound constant to 0.2083. Their approach utilizes the known values of B_n for up to n = 12. We tackle the lower bound by utilizing dynamic programming and the Lindström–Gessel–Viennot lemma. Our new bound is b_n ≥ 0.2721 for sufficiently large n. The result is based on a delicate interplay of theoretical ideas and computer assistance.
@InProceedings{corteskuhnast_et_al:LIPIcs.SoCG.2024.43, author = {Cort\'{e}s K\"{u}hnast, Fernando and Dallant, Justin and Felsner, Stefan and Scheucher, Manfred}, title = {{An Improved Lower Bound on the Number of Pseudoline Arrangements}}, booktitle = {40th International Symposium on Computational Geometry (SoCG 2024)}, pages = {43:1--43:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-316-4}, ISSN = {1868-8969}, year = {2024}, volume = {293}, editor = {Mulzer, Wolfgang and Phillips, Jeff M.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.43}, URN = {urn:nbn:de:0030-drops-199880}, doi = {10.4230/LIPIcs.SoCG.2024.43}, annote = {Keywords: counting, pseudoline arrangement, recursive construction, bipermutation, divide and conquer, dynamic programming, computer-assisted proof} }
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