Interval variety: Difference between revisions
Note the abbreviation SV |
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* 9 notes: none | * 9 notes: none | ||
* 10 notes: '''0010020302''' and '''0102103012''' | * 10 notes: '''0010020302''' and '''0102103012''' | ||
* 11 to | * 11 to 15 notes: none | ||
(Note that abstract SV4-ness implies that a scale pattern is [[primitive]], or single-period.) | (Note that abstract SV4-ness implies that a scale pattern is [[primitive]], or single-period.) | ||
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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. | ||