By Flajolet Ph., Sedgewick R.
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Extra info for Analytic combinatorics - symbolic combinatorics
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. Precisely, the decomposition based on the leftmost occurrence of ❮ in ñ⑥❮ is ❛ (37) ñ☛✵➠❴✙❮ ï❱è✼✵ ❞ ❼ ❴✳Õ ❩ ❿ ❑ Õ ❩ ❿ ý ❖❀❖▲❖ Õ ▼ ❛ ✆ ô➒õ✳➋ ö corresponding to the configurations ✷❷✷✁✷❷✷❷✷✁✷ ✷❷✷❷✷✁✷❷✷✁✷ ñ ï ❮ ✷✁✷❷✷❷✷✁✷❷✷ ✷❷✷✁✷❷✷✁✷❷✷ ÷ ❮ ø❻ù Õ ❩ ❿ ❑ ❖▲❖▲❖ Õ ▼ ú è The translation of the system (36), (37) into OGF’s then gives: The OGF of words not containing the pattern ❮ is ❝ û ❝ û➵ï (38) ➧ ❝ ▼ ö ð❤✡ ❣Õ ø ➆ ❝û ❝û✆ ø ø ◆äï✜❩ ➌ ✡ ❮➸ø ➌ the pattern length, and ❝ û the where ➆ is the alphabet cardinality, ✡ autocorrelation polynomial, ❝ û➵ï ò ❩ ❩ ❝ .
From the combinatorial standpoint, these examples illustrate the counting of structures that are richer than words (namely, pattern occurrences) by means of regular specifications. ✻ 22. Patterns with gaps. If less than ➀ symbols of the text must separate the letters of the pattern in order to form a valid occurrence, then the OGF of occurrences is ➛ ➭ See  for variations of this theme. ➳ ✽✪➙ ➳ ✽✪➙ ➂ ➥ ➥ ➛ ➂ ➸➭ ➝ ✮ ❁ ➛♣➸ ➭ ➧✠✮ ❃ I. 2. Finite automata. Let again a finite alphabet ➃ be fixed.
For instance, õ ③▼ ⑥ ❩ ï⑦❝❴ ➾ ✆ ➚ ✆ ✆ ❛ , there are 15✝ ways✞ to partition it (Figure 9). Let③ ▼ñ ⑥ if the domain is õ denote ③the ï ▼ ⑥ collection of all partitions of the set ð î into ◆ non–empty blocks and ➧ ✺❀✺ õ under consideration here is a card ñ object û the corresponding cardinality. The basic set partition (not to be confused with integer partitions considered earlier). ø ③▼ ⑥ It is possible to find an encoding of partitions❛ in ñ of an î –set into ◆ blocks by ▼ as follows: words over a ◆ letter alphabet, ❵ï❡❴ ❑ ý ➊ ★ ✆ ➊ ✆ ✺❀✺▲✺ ✤✆ ➊ ✤ ✥ ✦ I.
Analytic combinatorics - symbolic combinatorics by Flajolet Ph., Sedgewick R.