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By Flajolet P., Sedgewick R.

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6) = 1 + 2 + 3 + 6 = 12). 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 = .

Binary words. The OGF of binary words, as seen already, can be obtained directly from the iterative specification, 1 , W = S EQ(Z + Z) =⇒ W (z) = 1 − 2z whence the expected result, Wn = 2n . General trees. The recursive specification of general trees leads to an implicit definition of their OGF, z G = Z × S EQ(G) =⇒ G(z) = . 1 − G(z) From this point on, basic algebra does the rest. First the original equation is equivalent (in the ring of formal power series) to G − G2 − z = 0. Next, the quadratic equation I.

N−1 zk (1 − z)k k−1 ∞ Y (1 − z m )−1 — m=1 r Y —, sum. ≤ r MS ET(S EQ1 . r (Z)) —, ≤ k sum. ∼ = MS ET(S EQ1 . k (Z)) Cyclic comp. , distinct sum. PS ET(S EQ≥1 (Z)) m=1 k Y m=1 (1 − z m )−1 (1 − z m )−1 Eq. (35) ∞ Y (1 + z m ) m=1 — asympt. 1 n 2 2 cr ρ−n r nk−1 (k − 1)! (r − 1)! (k − 1)! 2n Eq. (36) n 33/4 π√ n3 — e 12n3/4 — F IGURE 8. Partitions and compositions: specifications, generating functions, counting sequences, and asymptotic approximation. ,k (z) of (32). This last relation provides a bona fide algorithm for computing the initial values of the number of stack polyominoes (EIS A001523): S(z) = z + 2 z 2 + 4 z 3 + 8 z 4 + 15 z 5 + 27 z 6 + 47 z 7 + 79 z 8 + · · · .

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