By Flajolet Ph., Sedgewick R.

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E XAMPLE 5. Patterns in a random text. A sequence of letters that occurs in the right order, but not necessarily contiguously in a text is said to be a “hidden pattern”. 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 ◆ .

It is an atom. A word is then any finite sequence of letters, usually written without separators. So, for us, with the choice of the latin alphabet ( ➃ ï ❛ ❴ a,. . ,z ), sequences written ✶ as ygololihq, philology, zgrmblglps are words. The set of all words (often written as ➃ in formal linguistics) will be consistently denoted by ➄ here. Following a well-established tradition in theoretical computer science and formal linguistics, any subset of ➄ is called a language (or formal language, when the distinction with natural languages has to be made).

The following equivalence theorem is briefly discussed in the Appendix (see A PPEN DIX : Regular languages, p. 171): T HEOREM (Kleene–Rabin–Scott). , recognizable by a deterministic finite automaø✾Ü ton); ø✾û Ü➒Üto be the set of words accepted by a nondeterministic finite automaton; û toø✾Ü➒Ü➒beÜ described by a standard regular expression. ø✾Ü➒Ý 34 I. UNLABELLED STRUCTURES AND ORDINARY GENERATING FUNCTIONS In the case of a deterministic automaton, it is easy to determine whether a word Ø is accepted: it suffices to start from the initial state ❹ ❼ , scan the letters of the word from left to right, and follow at each stage the only transition permitted; the word is accepted if the state reached in this way after scanning the last letter of Ø is a final state.

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


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