Analytic combinatorics MAc by Flajolet P., Sedgewick R.

By Flajolet P., Sedgewick R.

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Semantics of recursion. 9: Tree concepts, p. 736 for basic definitions). In graph theory, a tree is classically defined as an undirected graph that is connected and acyclic. Additionally, a tree is rooted if a particular vertex is distinguished (this vertex is then kown as the root). Computer scientists commonly make use of trees called plane3 that are rooted but also embedded in the plane, so that the ordering of 3The alternative terminology “planar tree” is also often used, but it is frowned upon by some as incorrect (all trees are planar graphs).

In precise terms, one has C YC(B) := S EQ(B)/S, where S is the equivalence relation between sequences defined by (β1 , . . , βr ) S (β1′ , . . , βr′ ) iff there exists some circular shift τ of [1 . r ] such that for all j, β ′j = βτ ( j) ; in other words, for some d, one has β ′j = β1+( j−1+d) mod r . Here is for instance a depiction of the cycles formed from the 8 and 16 sequences of lengths 3 and 4 over two types of objects (a, b): the number of cycles is 4 (for n = 3) and 6 (for n = 4). Sequences are grouped into equivalence classes according to the relation S: (17) 3–cycles : aaa aab aba baa abb bba bab , bbb  aaaa    aaab aaba abaa baaa aabb abba bbaa baab 4–cycles : .

5, p. 26) respectively defined by N = C YC(Z + Z) and I = S EQ≥1 (Z). From this, one can construct ever more complicated objects. For instance, P = MS ET(I) ≡ MS ET(S EQ≥1 (Z)) means the class of multisets of positive integers, which is isomorphic to the class of integer partitions (see Section I. 3 below for a detailed discussion). As such examples demonstrate, a specification that is iterative can be represented as a single term built on E, Z and the constructions +, ×, S EQ, C YC, MS ET, PS ET.

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