Analytic combinatorics by Flajolet P., Sedgewick R.

By Flajolet P., Sedgewick R.

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The partitions of n into odd summands (On ) and into distinct summands (Qn ) are equinumerous. Indeed, one has ∞ ∞ Y Y (1 − z 2j+1 )−1 . (1 + z m ), O(z) = Q(z) = j=0 m=1 2 Equality results from substituting (1 + a) = (1 − a )/(1 − a) with a = z m , 1 − z2 1 − z4 1 − z6 1 − z8 1 − z10 1 1 1 ··· = ··· , 1 − z 1 − z2 1 − z 3 1 − z4 1 − z 5 1 − z 1 − z3 1 − z5 and simplification of the numerators with half of the denominators (in boldface). Q(z) = 46 I. UNLABELLED STRUCTURES AND ORDINARY GENERATING FUNCTIONS Partitions into powers.

The technique is generally applicable to powersets and multisets; see Note √ 40 for another application. ) By varying (27) and (28), 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 OGF of the classes C T := S EQ(S EQ T (Z)) and P T := MS ET(S EQ T (Z)) of compositions and partitions having summands restricted to T is given by 1 1 1 = . , P T (z) = C T (z) = n 1 − n∈T z 1 − T (z) 1 − zn n∈T P ROOF.

They are named after P´olya who first developed the general enumerative theory of objects under permutation groups [36, 318, 320]. 2 signifies that iterative classes have explicit generating functions involving compositions of the basic operators only, while recursive structures have OGFs that are accessible indirectly via systems of functional equations. As we see at various places in this chapter, the following classes are constructible: binary words, binary trees, general trees, integer partitions, integer compositions, nonplane trees, polynomials over finite fields, necklaces, and wheels.

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