By Flajolet P., Sedgewick R.
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S EQ≥1 {•}, which constitutes a direct way to visualize the equality I (z) = z/(1 − z). Compositions. 1, a direct translation into OGF: 1 (33) C = S EQ(I) ⇒ C(z) = . 1 − I (z) The collection of equations (31), (33) thus fully determines C(z): C(z) = 1−z 1 z = 1 − 1−z 1 − 2z = 1 + z + 2z 2 + 4z 3 + 8z 4 + 16z 5 + 32z 6 + · · · . From here, the counting problem for compositions is solved by a straightforward expansion of the OGF: one has C(z) = n≥0 2n z n − n≥0 2n z n+1 , 40 I.
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 : . 1, p. 18). We make only a limited use of it for unlabelled objects; however, its counterpart plays a rather important rˆole in the context of labelled structures and exponential generating functions of Chapter II. Multiset construction. Following common mathematical terminology, multisets are like finite sets (that is the order between elements does not count), but arbitrary repetitions of elements are allowed.
250 (left), the number of compositions Cn (middle) and the number of partitions (right). The figure illustrates the difference in growth √ between Cn = 2n−1 and Pn = e O( n) . implying C0 = 1 and Cn = 2n − 2n−1 for n ≥ 1; that is, Cn = 2n−1 , n ≥ 1. 6), of which there are clearly 2n−1 possibilities. Partitions. 1, p. 26, provides (35) P = MS ET(I) ⇒ P(z) = exp I (z) + 1 1 I (z 2 ) + I (z 3 ) + · · · 2 3 , together with the product form corresponding to (22), p. 28, P(z) = (36) ∞ m=1 1 1 − zm = 1 + z + z2 + · · · 1 + z2 + z4 + · · · 1 + z3 + z6 + · · · · · · = 1 + z + 2z 2 + 3z 3 + 5z 4 + 7z 5 + 11z 6 + 15z 7 + 22z 8 + · · · (the counting sequence is EIS A000041).