By Flajolet P., Sedgewick R.

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A variant specification of triangulations. Consider the class U of “non-empty” triangulations of the n-gon, that is, we exclude the 2-gon and the corresponding “empty” triangulation of size 0. Then U = T \ {ǫ} admits the specification U = ∇ + (∇ × U) + (U × ∇) + (U × ∇ × U) which also leads to the Catalan numbers via U = z(1 + U )2 , so that U (z) = (1 − 2z − √ 1 − 4z)/(2z) ≡ T (z) − 1. ✁ I. 4. Exploiting generating functions and counting sequences. In this book we are going to see altogether more than a hundred applications of the symbolic method.

For all the constructions of this book, there furthermore exists a function ϑ(n), such that (An ) only (1) (m) depends on the first ϑ(n) elements of the (Bk ), . . , (Bk ), with ϑ(n) ≤ K n + L (H¨older condition) or ϑ(n) ≤ n + L (Lipschitz condition). For instance, the functional f (z) → f (z 2 ) is H¨older; the functional f (z) → ∂z f (z) is Lipschitz. ✁ I. 2. Admissible constructions and specifications The main goal of this section is to introduce formally the basic constructions that constitute the core of a specification language for combinatorial structures.

5. The Catalan numbers Cn , their Stirling approximation Cn⋆ = 4n / π n 3 , and the ratio Cn⋆ /Cn . 12. Experimental asymptotics. 5, guess the values6 of C10 7 107 and of C ⋆ 6 /C5·106 to 25D. ) ✁ I. 3. Integer compositions and partitions This section and the next few provide examples of counting via specifications in classical areas of combinatorial theory. They illustrate the benefits of the symbolic method: generating functions are obtained with hardly any computation, and at the same time, many counting refinements follow from a basic combinatorial construction.

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Analytic combinatorics MAc by Flajolet P., Sedgewick R.

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