Analytic combinatorics - symbolic combinatorics by Flajolet Ph., Sedgewick R.

By Flajolet Ph., Sedgewick R.

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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. 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 ❩ ❩ ❝ .

The book of van Rensburg [144] describes many such constructions and their relation to certain models of statistical physics. ❃ ❨ ➳ ➛♣➸ ❰ Ý I. 2. Integer related constructions. Finally, we say a few words about the two constructions of cycle and powerset that haven’t been yet applied to ❫ . First, the class ❩ ï ● ❴➲❜ ❛ comprises cyclic compositions, that is, compositions defined up to circular shift; so, for instance ö ö✐ ❩ ✈ð♠ö ö ✈ , ✐ ö✈ð♠ö ö ✈ ö , etc, are identified. Alternatively, we may view elements composed ú of as “wheels” ú ú of ú circular arrangements of segments (taken up to circular symmetry).

20. Runs in arbitrary alphabets. For an alphabet of cardinality ➥ , the quantity ✽✰➙ ➥ ✽✰➙➵➛ ➭ ➛♥➦ ➳ ➙➠✽✛➸★➛ ➭ ➥ ➧ ✮ ❈ is the OGF of words without ➫ consecutive occurrences of a designated letter. ❃ The case of longest runs exemplifies the expressive power of nested constructions involving sequences. 7. , letters of a finite alphabet ➃ ) together with combinatorial sums, cartesian products, and sequence constructions is said to be a regular specification. A language ➦ is said to be ➧ -regular (specification-regular) if there exists a regular specification ➨ such that ➦ and ➨ are combinatorially isomorphic, ➦ ïþ ➨ .

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