Analytic combinatorics by Flajolet P., Sedgewick R. PDF

By Flajolet P., Sedgewick R.

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Geometrically, partitions, are represented as collections of points: this is the staircase model of Figure 6). A symmetry around the main diagonal (also known in the specialized literature as conjugation) exchanges number of summands and value of largest summand: one has (with previous notations) {1, . k} P (≤k) ∼ =P =⇒ P (≤k) (z) = P {1, . ,k} = k Y m=1 1 . 1 − zm As a consequence, the OGF of partitions with exactly k summands, P (k) (z) = P (≤k) (z) − P (≤k−1) (z), evaluates to P (k) (z) = zk .

A specification for an r–tuple A = (A(1) , . . , A(r) ) of classes is a collection of r equations,  (1) Ξ1 (A(1) , . . , A(r) )   A(2) = A = Ξ2 (A(1) , . . , A(r) ) (21)   (r) · · · A = Ξr (A(1) , . . , A(r) ) where each Ξi denotes a term built from the A’s using the constructions of disjoint union, cartesian product, sequence, set, multiset, and cycle, as well as the initial structures E and Z. We also say that the system is a specification of A(1) . A specification for a class of combinatorial structures is thus a sort of formal grammar defining that class.

Find one such bijection. [Hint: the construction of triangulations is evocative of binary trees, and binary trees are themselves in bijective correspondence with general trees; see A PPENDIX A: Tree concepts, p. 625]. 9. A variant specification of triangulations. Consider the class U of “nonempty” 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 and U (z) = (1 − 2z − √ 1 − 4z)/(2z), so that U (z) = T (z) − 1.

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Analytic combinatorics by Flajolet P., Sedgewick R.


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