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By Flajolet Ph., Sedgewick R.

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This famous identity expresses Û as ❇ ✟ ➪ ➳ ✽✪➙➵➛ ➸ Ý ➳ ➙❤✽✓➸ ➭ ➛ ❻➭ ❽ ➭ ❁ ➧✠❿✮ ❾ ✴ ❰ ✟ ➔ ➭ ❻ ❝ ❺ ❼ ✮ Ò Ó å It is proved formally and combinatorially in [28, p. 105]. As a consequence, the numbers ➒▲Û Þ ↔ Þ ➹ ✭ ➳äç ç ➸ can be determined in æ arithmetic operations. ❃ ➴ I. 4. WORDS AND REGULAR LANGUAGES 29 ✻ 19. Lattice points. ) I. 4. Words and regular languages First a finite alphabet ➃ whose elements are called letters is fixed. , it is an atom. A word is then any finite sequence of letters, usually written without separators.

Integer compositions and partitions This section and the next one provide first illustrations of the symbolic method and of counting via specifications. In this framework, generating functions are obtained with hardly any computation. At the same time, many counting refinements follow from a basic combinatorial construction. The most direct applications described here relate to the additive decomposition of integers into summands with the classical combinatorialarithmetic structures of partitions and compositions.

A popular denumerant problem consists in finding the number of ways of giving change of 99 cents using coins that are pennies (1 /c), nickels (5 /c), dimes (10 /c) and quarters (25 õ /c). ) For the case of a finite è , we predict from Proposition 2 that ① ë ❝ û is always a rational function with poles that are at roots of unity; also the ① ë satisfy a linear ø recurrence related to the structure of è . The solution to the original coin change problem is found to be ✄ ✝ ❝ ➐✳➐ ✞ ð ❑❼ ï ❝ ❝ ❝ ❝ ❸ ❸ ð❤❣ û ð❤❣ û ð❤❣ û ð ❣ ý û ú ú✺ õ õ ø p.

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