By Flajolet Ph., Sedgewick R.
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Submit yr notice: First released January 1st 2004
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Extra resources for Analytic combinatorics - symbolic combinatorics
WORDS AND REGULAR LANGUAGES 37 ï ð ✆ ò ✆ ò ✆ ò ✆ ð ✆ ð➯û . Theõ autocorrelation polynomial is defined ▼ ❑ ø ❝ ➤ û ì ï ❘❞ ❼ð✡ ❘▲❝ ✡ ✺ ➋ ø ❝ ✁ ❝ ❶ ➍ ❝ For the example pattern, this gives û➵ï✦ð ö ö ❸ . ✡ Let ñ be the language of words ø with no occurrence of ❮ and è the language of words The autocorrelation is then ✡ as that end with ❮ but have no other occurrence of ❮ . First, by appending a letter to a word of ñ , one finds a nonempty word either in ñ or è , so that ❛ ñ➀ö⑩❶ è ï❡❝❴ ì öòñ☛✵ó➃ (36) ✺ Next, appending a copy of the word ❮ to a word in ñ may only give words that contain ❮ at or “near” the end.
For instance the pattern “combinatorics” is to be found hidden in Shakespeare’s Hamlet (Act I, Scene 1) comb at in which our v a lian t Hamlet [. . ] f or fe i t [. . ] Whi c h he s tood . . ✺ Is this the sign of a secret encouragement passed to us by the author of Hamlet? ✉ for English). Let ❮❹ï Take a fixed finite alphabet ➃ comprising ➆ letters (➆ ï Õ ❑ Õ ý ❖❀❖▲❖äÕ ▼ be a word of length ◆ . Consider the regular specification ú ❰ ï îs❴❝➃ ❛ Õ ❑ îs❴❝➃ ❛ Õ ý îs❴❝➃ ❛ ❖▲❖❀❖✳î ❴✫➃ ❛ Õ ▼ ❑ îs❴✫➃ ❛ Õ ▼ îs❴❝➃ ❛ ✺ ❰ ◆ ö✦ð➈û -tuple whose first component is an arbitrary word, whose An element of is a ✪ second component is the so on, with letters of the pattern and free blocks ø❲ú letter Õ ❑ , and ❰ alternating .
Let again a finite alphabet ➃ be fixed. We first define a simple device that is able to “process” words over the alphabet and has wide descriptive power as regards structural properties of words. 8. A finite automaton is a directed multigraph whose edges are labelled õ to call the by letters of the alphabet. It is customary vertices by the name of✐×Ö states✐ and ✐ ✐ denote by the set of states. An initial state ❹ ❼ Ð and a set of final states are é õ by the automaton if there✐×Öexists a path also designated.
Analytic combinatorics - symbolic combinatorics by Flajolet Ph., Sedgewick R.