By Flajolet P., Sedgewick R.
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Extra info for Analytic combinatorics
43 (p. 71) for another application. 19 (p. ) ✁ By varying (33) and (35), we can use the symbolic method to derive a number of counting results in a straightforward manner. 1. Let T ⊆ I be a subset of the positive integers. The OGFs of the classes C T := S EQ(S EQT (Z)) and P T := MS ET(S EQT (Z)) of compositions and partitions having summands restricted to T ⊂ Z≥1 are given by C T (z) = 1 1− n∈T zn = 1 , 1 − T (z) P T (z) = n∈T 1 . 1 − zn Proof. 1, p. 26. This proposition permits us to enumerate compositions and partitions with restricted summands, as well as with a fixed number of parts.
1, p. 26, provides (35) P = MS ET(I) ⇒ P(z) = exp I (z) + 1 1 I (z 2 ) + I (z 3 ) + · · · 2 3 , together with the product form corresponding to (22), p. 28, P(z) = (36) ∞ m=1 1 1 − zm = 1 + z + z2 + · · · 1 + z2 + z4 + · · · 1 + z3 + z6 + · · · · · · = 1 + z + 2z 2 + 3z 3 + 5z 4 + 7z 5 + 11z 6 + 15z 7 + 22z 8 + · · · (the counting sequence is EIS A000041). Contrary to compositions that are counted by the explicit formula 2n−1 , no simple form exists for Pn . Asymptotic analysis of the OGF (35) based on the saddle-point method (Chapter VIII, p.
Coefficient extraction. We let generally [z n ] f (z) denote the operation of extracting the coefficient of z n in the formal power series f (z) = f n z n , so that (6) (The coefficient extractor [z n ] [z n ] f (z) n≥0 fn zn = fn . ) 20 I. 2. A molecule, methylpyrrolidinyl-pyridine (nicotine), is a complex assembly whose description can be reduced to a single formula corresponding here to a total of 26 atoms. The OGFs corresponding to our three examples W, P, T are then ∞ 1 W (z) = 2n z n = 1 − 2z n=0 ∞ P(z) = n!
Analytic combinatorics by Flajolet P., Sedgewick R.