Glossary for combinatorics on words: Difference between revisions

Line 17:

|-

| factor, subword || || ''u'' is a ''factor'' of ''w'' if ''w = yuv'' for words ''y'' and ''v''.

|-

| primitive || single-period || ''w'' is ''primitive'' if for all ''u'' and all ''m'' ≥ 2, ''u''m ≠ ''w''.

|-

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

|-

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

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

|-

| Sturmian word (Note: definitions may vary.) || aperiodic MOS scale || A binary cutting word where the line has irrational slope.

|-

| 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~~.

| 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''.

|-

| spectrum<ref>Bulgakova, D. V., Buzhinsky, N., & Goncharov, Y. O. (2023). On balanced and abelian properties of circular words over a ternary alphabet. Theoretical Computer Science, 939, 227-236.</ref> || [[interval class]] ||

Line 30:

Line 32:

| abelian complexity || [[interval variety|variety]] of an [[interval class]] ||

|-

| Parikh vector || interval occurring in a scale || A given subword ''w'' is associated with a ''Parikh vector'' whose coefficient for each letter ''a'' is |''w''|''a''. The Parikh vector of a length-''k'' subword is a ''k''-step.

| Parikh vector || interval occurring in a scale || A given subword ''w'' is associated with a ''Parikh vector'' whose coefficient for each letter ''a'' is |''w''|''a'', the number of occurrences of ''a'' in ''w''. The Parikh vector of a length-''k'' subword is a ''k''-step.

|-

| (1-)balanced word || (for binary words) MOS scale || A word such that for any k, the number of occurrences of any letter in any two k-steps differ by at most 1.

| (1-)balanced word || (for binary words) MOS scale || A word such that for any ''k'', the number of occurrences of any letter in any two ''k''-steps differ by at most 1.

|}

== References ==

@@ Line 17: / Line 17: @@
 |-
 | factor, subword || || ''u'' is a ''factor'' of ''w'' if ''w = yuv'' for words ''y'' and ''v''.
+|-
+| primitive || single-period || ''w'' is ''primitive'' if for all ''u'' and all ''m'' &ge; 2, ''u''<sup>m</sup> &ne; ''w''.
 |-
 | Christoffel word || brightest mode of a periodic [[MOS scale]] ||
 |-
-| Lyndon word || lexicographically brightest mode || A word that is lexicographically first among its rotations
+| Lyndon word || lexicographically brightest mode || A word that is lexicographically first among its rotations.
 |-
 | Sturmian word (Note: definitions may vary.) || aperiodic MOS scale || A binary cutting word where the line has irrational slope.
 |-
-| 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.
+| 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>.
 |-
 | spectrum<ref>Bulgakova, D. V., Buzhinsky, N., & Goncharov, Y. O. (2023). On balanced and abelian properties of circular words over a ternary alphabet. Theoretical Computer Science, 939, 227-236.</ref> || [[interval class]] ||
@@ Line 30: / Line 32: @@
 | abelian complexity || [[interval variety|variety]] of an [[interval class]] ||
 |-
-| Parikh vector || 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 Parikh vector of a length-''k'' subword is a ''k''-step.
+| Parikh vector || 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.
 |-
-| (1-)balanced word || (for binary words) MOS scale || A word such that for any k, the number of occurrences of any letter in any two k-steps differ by at most 1.
+| (1-)balanced word || (for binary words) MOS scale || A word such that for any ''k'', the number of occurrences of any letter in any two ''k''-steps differ by at most 1.
 |}
 == References ==