Download PDF by Flajolet P., Sedgewick R.: Analytic combinatorics

By Flajolet P., Sedgewick R.

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In order to enumerate the class C {1,2} of compositions of n whose parts are only allowed to be taken from the set {1, 2}, simply write C {1,2} = S EQ(I {1,2} ) with I {1,2} = {1, 2}. Thus, in terms of generating functions, one has C {1,2} (z) = 1 1 − I {1,2} (z) with I {1,2} (z) = z + z 2 . This formula implies C {1,2} (z) = 1 = 1 + z + 2z 2 + 3z 3 + 5z 4 + 8z 5 + 13z 6 + · · · , 1 − z − z2 and the number of compositions of n in this class is expressed by a Fibonacci number, √ n √ n 1 1+ 5 1− 5 {1,2} Cn = Fn+1 where Fn = √ − , 2 2 5 of daisy–artichoke–rabbit fame In particular, the rate of growth is of the exponential type ϕ n , √ 1+ 5 is the golden ratio.

5 exemplifies the quality of the approximation with subtler phenomena also apparent on the figures and well explained by asymptotic theory. Such asymptotic formulae then make comparison between the growth rates of sequences easy. The interplay between combinatorial structure and asymptotic structure is indeed the principal theme of this book. We shall see in Part B that the generating functions provided by the symbolic method typically admit similarly simple asymptotic coefficient estimates. 11.

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 by Flajolet P., Sedgewick R.

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