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(Scales are understood to be abstract with equaves unspecified.)
(Scales are understood to be abstract with equaves unspecified.)
Also check the main [[Glossary]].


{| class="wikitable sortable"
{| class="wikitable sortable"
|+ Definitions
|-
|-
! Academic term(s) !! Xen term(s) !! Definition
! Academic term(s) !! Xen term(s) !! Brief definition
|-
|-
| alphabet || steps || A countable set of symbols called letters.
| alphabet || steps || A countable set of symbols called letters.
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| conjugate || equivalent under modal rotation || Words ''u'' and ''v'' are conjugate if there exist words ''s'' and ''t'' such that ''u'' = ''st'' and ''v'' = ''ts''. Compare: any two group elements of the form ''st'' and ''ts'' are also conjugate.
| conjugate || equivalent under modal rotation || Words ''u'' and ''v'' are conjugate if there exist words ''s'' and ''t'' such that ''u'' = ''st'' and ''v'' = ''ts''. Compare: any two group elements of the form ''st'' and ''ts'' are also conjugate.
|-
|-
| circular word, necklace || [[periodic scale]] || An equivalence class of words that are conjugate, or equivalently, an infinite periodic word.
| circular word, necklace || [[periodic scale]] || An equivalence class of words that are conjugate.
|-
|-
| factor, subword || || ''u'' is a ''factor'' of ''w'' if ''w = yuv'' for words ''y'' and ''v''. ''u'' is a factor of a circular word [''w''] if it is a factor of some representative of [''w''].
| factor, subword || || ''u'' is a ''factor'' of ''w'' if ''w = yuv'' for words ''y'' and ''v''. ''u'' is a factor of a circular word [''w''] if it is a factor of some representative of [''w''].
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| episturmian word || || An infinite word ''s'' is ''episturmian'' provided that 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"/>
| episturmian word || || An infinite word ''s'' is ''episturmian'' provided that 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, assuming an ordering on the letters.
| [[Lyndon word]] || lexicographically brightest mode || A word that is lexicographically first among its rotations, assuming an ordering on the letters.
|-
|-
| 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>.
| 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>.
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| abelian complexity || (of the set of ''k''-steps) [[interval variety|variety]] of an [[interval class]]||  
| abelian complexity || (of the set of ''k''-steps) [[interval variety|variety]] of an [[interval class]]||  
|-
|-
| Parikh vector; abelianization || interval occurring in a scale || A given subword ''w'' is associated with a ''Parikh vector'' whose coefficient for each letter ''a'' is &#124;''w''&#124;<sub>''a''</sub>, the number of occurrences of ''a'' in ''w''. The Parikh vector of a length-''k'' subword is a ''k''-step in xen terminology.
| Parikh vector; abelianization || interval occurring in a scale; step decomposition || A given subword ''w'' is associated with a ''Parikh vector'' whose coefficient for each letter ''a'' is &#124;''w''&#124;<sub>''a''</sub>, the number of occurrences of ''a'' in ''w''. The Parikh vector of a length-''k'' subword is a ''k''-step in xen terminology.
|-
|-
| (1-)[[balance]]d word || (for binary words) MOS scale || A word such that for any ''k'' and any two ''k''-steps, the numbers of occurrences of any letter differ by at most 1.
| (1-)[[balance]]d word || (for binary words) MOS scale || A word such that for any ''k'' and any two ''k''-steps, the numbers of occurrences of any letter differ by at most 1.
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== References ==
== References ==
[[Category:Terms]]
[[Category:Terms| ]]
[[Category:Scale]]
[[Category:Scale]]
[[Category:Combinatorics on words]]
[[Category:Combinatorics on words]]
Retrieved from "https://en.xen.wiki/w/Glossary_for_combinatorics_on_words"