By Flajolet P., Sedgewick R.

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The translation of the cycle relation A = C YC (B) turns out to be ∞ ϕ(k) 1 A(z) = log , k 1 − B(z k ) = exp k=1 where ϕ(k) is the Euler totient function. The first terms, with Lk (z) := log(1 − B(z k ))−1 are 1 1 2 2 4 2 A(z) = L1 (z) + L2 (z) + L3 (z) + L4 (z) + L5 (z) + L6 (z) + · · · . 1 2 3 4 5 6 We defer the proof to A PPENDIX A: Cycle construction, p. 618, since it relies in part on multivariate generating functions to be officially introduced in Chapter III. The results for sets, multisets, and cycles are particular cases of the well known P´olya theory that deals more generally with the enumeration of objects under group symmetry actions [318, 320].

Consequently, P1 , . . , PN can be computed in O(N 2 ) integer-arithmetic operations. (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 = .

4 12 There ⌈x⌋ ≡ ⌊x + 21 ⌋ denotes the integer closest to the real number x. Such results are typically obtained by the two step process: (i) decompose the rational generating function into simple fractions; (ii) compute the coefficients of each simple fraction and combine them to get the final result [76, p. 108]. The general argument also gives the generating function of partitions whose summands lie in the set {1, 2, . . ,r} (z) = r Y m=1 1 . 1 − zm In other words, we are enumerating partitions according to the value of the largest summand.

### Analytic combinatorics by Flajolet P., Sedgewick R.

by Ronald

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