By Flajolet P., Sedgewick R.
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A specification for an r –tuple A = (A(1) , . . , A(r ) ) of classes is a collection of r equations, (1) (1) (r ) 1 (A , . . , A ) A(2) = (1) (r ) A = 2 (A , . . , A ) (26) · · · (r ) (1) (r ) A = r (A , . . , A ) where each i denotes a term built from the A using the constructions of disjoint union, cartesian product, sequence, set, multiset, and cycle, as well as the initial classes E (neutral) and Z (atomic). We also say that the system is a specification of A(1) . A specification for a combinatorial class is thus a sort of formal grammar defining that class.
A ) where each i denotes a term built from the A using the constructions of disjoint union, cartesian product, sequence, set, multiset, and cycle, as well as the initial classes E (neutral) and Z (atomic). We also say that the system is a specification of A(1) . A specification for a combinatorial class is thus a sort of formal grammar defining that class. Formally, the system (26) is an iterative or non-recursive specification if it is strictly upper-triangular, that is, A(r ) is defined solely in terms of initial classes Z, E; the definition of A(r −1) only involves A(r ) , and so on; in that case, by back substitutions, it is apparent that for an iterative specification, A(1) can be equivalently described by a single term involving only the initial classes and the basic constructors.
The precise way of defining MS ET(B) is as a quotient: MS ET(B) := S EQ(B)/R with R, the equivalence relation of sequences being defined by (α1 , . . , αr ) R (β1 , . . , βr ) iff there exists some arbitrary permutation σ of [1 . r ] such that for all j, β j = ασ ( j) . Powerset construction. The powerset class (or set class) A = PS ET(B) is defined as the class consisting of all finite subsets of class B, or equivalently, as the class PS ET(B) ⊂ MS ET(B) formed of multisets that involve no repetitions.
Analytic combinatorics MAc by Flajolet P., Sedgewick R.
by Paul
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