Analytic combinatorics MAc by Flajolet P., Sedgewick R.

By Flajolet P., Sedgewick R.

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At the start, the triangle that contains the two smallest labels). Then, a triangulation decomposes into its root triangle and two subtriangulations (that may well be “empty”) appearing on the left and right sides of the root triangle; the decomposition is illustrated by the following diagram: = + The class T of all triangulations can be specified recursively as T = {ǫ} + (T × ∇ × T ) , I. 2. ADMISSIBLE CONSTRUCTIONS AND SPECIFICATIONS 35 provided that we agree to consider a 2-gon (a segment) as giving rise to an “empty” triangulation of size 0.

28 I. COMBINATORIAL STRUCTURES AND ORDINARY GENERATING FUNCTIONS The exp–log transformation A(z) = exp(log A(z)) then yields (20) ∞ Bn log(1 + z n ) A(z) = exp = exp = B(z) B(z 2 ) B(z 3 ) exp − + − ··· , 1 2 3 n=1 ∞ n=1 Bn · ∞ (−1)k−1 k=1 z nk k where the second line results from expanding the logarithm, u2 u3 u − + − ··· , 1 2 3 and the third line results from exchanging the order of summation. The proof finally extends to the case of B being infinite by noting that each An depends only on those B j for which j ≤ n, to which the relations given above for the (≤m) = PS ET(B (≤m) ).

COMBINATORIAL STRUCTURES AND ORDINARY GENERATING FUNCTIONS (see also an earlier book by Sloane and Plouffe [523]) and contains more than 100,000 sequences. Indeed, the three sequences (Wn ), (Pn ), and (Tn ) are respectively identified2 as EIS A000079, EIS A000142, and EIS A000108. 1. Necklaces. How many different types of necklace designs can you form with n beads, each having one of two colours, ◦ and •, where it is postulated that orientation matters? Here are the possibilities for n = 1, 2, 3, .

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