Interval variety: Difference between revisions

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In the following, two letters are to be considered the same if their numerical values are congruent modulo ''n''.  
In the following, two letters are to be considered the same if their numerical values are congruent modulo ''n''.  


{{theorem|text=For all ''n'' ≥ 1, the word '''0123'''...('''''n''-1''') is SV''n''.}}
{{theorem|contents=For all ''n'' ≥ 1, the word '''0123'''...('''''n''-1''') is SV''n''.}}


{{Proof|contents=All ''k''-letter subwords of '''0123'''...('''''n''-1''') is of the form ('''i''')('''i+1''')...('''i+k-1'''), and there are exactly ''n'' of them.}}
{{Proof|contents=All ''k''-letter subwords of '''0123'''...('''''n''-1''') is of the form ('''i''')('''i+1''')...('''i+k-1'''), and there are exactly ''n'' of them.}}


{{theorem|text=For all ''n'' ≥ 1, the word '''0123'''...('''''n''-2''')('''''n''-1''')('''''n''-2''')...'''3210''' is SV''n''.}}
{{theorem|contents=For all ''n'' ≥ 1, the word '''0123'''...('''''n''-2''')('''''n''-1''')('''''n''-2''')...'''3210''' is SV''n''.}}


{{Proof|We prove this by dividing this word into four overlapping noncircular subwords which cover all cases.  
{{Proof|We prove this by dividing this word into four overlapping noncircular subwords which cover all cases.