Recently, I got interested in the study of the combinatorial aspects of continued fractions. Precisely, I read of the following lemma of Flajolet (see [here](http://www.stat.purdue.edu/~mdw/ChapterIntroductions/ContinuedFractionsUpdateViennot.pdf)):


**Lemma.** It holds
$$\sum\_{\omega} \nu(\omega) \, z^{|\omega|} = \frac1{1 - c\_0 z - \displaystyle\frac{a\_0b\_0z^2}{1 - c\_1 z - \displaystyle\frac{a\_1 b\_1 z^2}{1 - c\_2 z - \ldots}}} ,$$
where $\omega$ runs over all the Motzkin paths, $|\omega|$ is the length of the Motzkin path $\omega$, and $\nu(\omega)$ is its weight, assigning the weights $a\_i$, $b\_i$, and $c\_i$ to up, down, and horizontal steps, respectively (see the linked PDF for more details).


I would like to know more about this kind of connections between combinatorics and continued fractions and **I am looking for a book about this subject**, or at least with a detailed chapter about.


Until know I just found some articles, all pointing to the 1980 article of Flajolet (1). I also see that the book (2) has a chapter on "Combinatorial interpretations of continued fractions", but it regards tilings and continued fractions with integers numerators and denominators, so it is about other things. The lecture notes of Viennot (3) might be good, but I cannot read French and they are only 3 years after Flajolet paper, it would be better something more updated.


Thank you in advance for your help.


(1) P. Flajolet, Combinatorial aspects of continued fractions, Discrete Mathematics 32 (1980) 125--161.


(2) A. T. Benjamin, J. J. Quinn, Proofs that Really Count: The Art of Combinatorial Proof, Mathematical Association of America, 2003.


(3) Une theorie combinatoire des polynomes orthogonaux generaux <http://www.xavierviennot.org/xavier/polynomes_orthogonaux.html>



Combinatorial aspects of continued fractions
One book that I know which covers this topic is (available online):
[Analytic Combinatorics](http://algo.inria.fr/flajolet/Publications/book.pdf), P. Flajolet and R. Sedgewick. In particular, (as the OP now also notes in a comment), the relevant section is V.4.


The [following slides](http://algo.inria.fr/flajolet/Publications/Slides/Leuven09.pdf) of Flajolet are also relevant.




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**Remarks**


1. The OP may also find [this paper](http://www.emis.ams.org/journals/EJC/Volume_7/PDF/v7i1r17.pdf) on restricted permutations and continued fractions of interest.
2. Another interesting reference (not on the original topic, but I thought it worthwhile to mention here) is: [Analytic Combinatorics in Several Variables](https://www.math.upenn.edu/~pemantle/papers/ACSV.pdf), by R. Pemantle and M. Wilson.


