By Flajolet Ph., Sedgewick R.

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**Extra info for Analytic combinatorics - symbolic combinatorics**

**Example text**

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 ➨ .

The it is based on an fundamental notion is that of an autocorrelation vector. For a given ❮ , this vector of bits ï ✡ ❼ ✆ ✳✆ ✡ ▼ ❑ û is most conveniently defined in terms of Iverson’s bracket as ✡ ø ✺▲✺❀✺ ✝ ✝ ❩ ï Õ ❑ Õ ❖▲❖❀❖❅Õ ▼ ❩ ï Õ ❩ ❿ ❑ Õ ❩ ❿ ❖▲❖❀❖❅Õ ▼ ✞ ✞ ý ý ✡ ✺ ❩ In other words, the bit is determined by shifting ❮ right by positions and putting a 1 if ✡ the remaining letters match the original. For instance, with ❮✪ï➴ ➉➋➉ , one has Ü ➉➋➉ a a b b a a _______________________ a a b b a a a a b b a a a a b b a a a a b b a a a a b b a a a a b b a a ➊❻➊ 1 0 0 0 1 1 I.

Identify each block by its smallest element called the block leader; then sort the block leaders into increasing order. Define the index of a block as the rank of its leader amongst all the ◆ leaders, with ranks conventionally starting at ð .

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

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