Interval variety: Difference between revisions
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== Open questions == | == Open questions == | ||
* Why are (abstractly) SV4 scale patterns seemingly so rare? | * Why are (abstractly) SV4 scale patterns seemingly so rare? | ||
** Conjecture: There are only finitely many SV4/MV4 [[circular word]]s. | ** Conjecture: There are only finitely many SV4/MV4 quaternary [[circular word]]s. | ||
** Conjecture: For all ''n'' greater than a sufficiently large ''m'', the longest abstractly SV-''n'' word is '''0123'''…({{nowrap| '''''n'' − 2''' }})({{nowrap| '''''n'' − 1''' }})({{nowrap| '''''n'' − 2''' }})…'''3210''', with length {{nowrap| 2''n'' − 1 }}. | ** Conjecture: For all ''n'' greater than a sufficiently large ''m'', the longest abstractly SV-''n'' word is '''0123'''…({{nowrap| '''''n'' − 2''' }})({{nowrap| '''''n'' − 1''' }})({{nowrap| '''''n'' − 2''' }})…'''3210''', with length {{nowrap| 2''n'' − 1 }}. | ||
** The following conjecture may be key to proving the ones above: For a sufficiently long [[arity|ternary]] noncircular word, there exists {{nowrap| ''k'' > 1 }} such that the interval class of ''k''-steps has at least 3 sizes and the interval class of {{nowrap|(''k'' − 1)}}-steps also has at least 3 sizes. | ** The following conjecture may be key to proving the ones above: For a sufficiently long [[arity|ternary]] noncircular word, there exists {{nowrap| ''k'' > 1 }} such that the interval class of ''k''-steps has at least 3 sizes and the interval class of {{nowrap|(''k'' − 1)}}-steps also has at least 3 sizes. | ||