Glossary for combinatorics on words: Difference between revisions

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| primitive || single-period || ''w'' is ''primitive'' if for all ''u'' and all ''m'' ≥ 2, ''u''''m'' ≠ ''w''. A circular word is primitive if one (thus any) representative word of it is primitive.

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| Christoffel word || brightest mode of a periodic [[MOS scale]] ||

| right (left) special factor || || A factor of ''w'' is ''right (left) special'' in ''s'' if there exist distinct letters ''x, y'' such that both ''ux'' and ''uy'' (resp. ''xu'' and ''yu'') are factors of ''s''.

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| Sturmian word || aperiodic MOS scale || An infinite binary word which has exactly (''n'' + 1) distinct length-''n'' subwords for every n ≥ 1.

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| Christoffel word || brightest mode of a periodic [[MOS scale]] || If ''p'' and ''q'' are relatively prime positive integers, then the ''Christoffel word'' of slope ''p''/''q'' is a word ''w'' of length ''p'' + ''q'' defined by ''w''[i] = ''x'' if ''ip'' mod ''n'' > (''i'' − 1)''p'' mod ''n'', ''y'' otherwise.<ref name="paquin">Geneviève Paquin, On a generalization of Christoffel words: epichristoffel words, Theoretical Computer Science, Volume 410, Issues 38–40, 2009, Pages 3782-3791, ISSN 0304-3975.</ref>

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| episturmian word || || An infinite word ''s'' is ''episturmian'' provided that if the set of the factors of ''s'' is closed under reversal and ''s'' has at most one right (equivalently left) special factor of each length.<ref name="paquin"/>

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| Lyndon word || lexicographically brightest mode || A word that is lexicographically first among its rotations.

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~~| Sturmian word (Note: Definitions may vary.) || aperiodic MOS scale || An infinite binary word which has exactly (''n'' + 1) distinct length-''n'' subwords for every n ≥ 1.~~

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| cutting word, cutting sequence || [[billiard scale]] || The word of letters given by traversing a line of a given direction, where each letter ''c''''i'' is an intersection of the line with the coordinate plane ''x''''i'' = ''m''''i''.

@@ Line 24: / Line 24: @@
 | primitive || single-period || ''w'' is ''primitive'' if for all ''u'' and all ''m'' &ge; 2, ''u''<sup>''m''</sup> &ne; ''w''. A circular word is primitive if one (thus any) representative word of it is primitive.
 |-
-| Christoffel word || brightest mode of a periodic [[MOS scale]] ||
+| right (left) special factor || || A factor of ''w'' is ''right (left) special'' in ''s'' if there exist distinct letters ''x, y'' such that both ''ux'' and ''uy'' (resp. ''xu'' and ''yu'') are factors of ''s''.
+|-
+| Sturmian word || aperiodic MOS scale || An infinite binary word which has exactly (''n'' + 1) distinct length-''n'' subwords for every n &ge; 1.
+|-
+| Christoffel word || brightest mode of a periodic [[MOS scale]] || If ''p'' and ''q'' are relatively prime positive integers, then the ''Christoffel word'' of slope ''p''/''q'' is a word ''w'' of length ''p'' + ''q'' defined by ''w''[i] = ''x'' if ''ip'' mod ''n'' > (''i'' − 1)''p'' mod ''n'', ''y'' otherwise.<ref name="paquin">Geneviève Paquin, On a generalization of Christoffel words: epichristoffel words, Theoretical Computer Science, Volume 410, Issues 38–40, 2009, Pages 3782-3791, ISSN 0304-3975.</ref>
+|-
+| episturmian word || || An infinite word ''s'' is ''episturmian'' provided that if the set of the factors of ''s'' is closed under reversal and ''s'' has at most one right (equivalently left) special factor of each length.<ref name="paquin"/>
 |-
 | Lyndon word || lexicographically brightest mode || A word that is lexicographically first among its rotations.
-|-
-| Sturmian word (Note: Definitions may vary.) || aperiodic MOS scale || An infinite binary word which has exactly (''n'' + 1) distinct length-''n'' subwords for every n &ge; 1.
 |-
 | cutting word, cutting sequence || [[billiard scale]] || The word of letters given by traversing a line of a given direction, where each letter ''c''<sub>''i''</sub> is an intersection of the line with the coordinate plane ''x''<sub>''i''</sub> = ''m''<sub>''i''</sub>.