Analytic combinatorics by Flajolet P., Sedgewick R.

By Flajolet P., Sedgewick R.

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From the definition of size for sums and products, there results that the size of a sequence is to be taken as the sum of the sizes of its components: γ = (α1 , . . , αℓ ) =⇒ |γ| = |α1 | + · · · + |αℓ |. Cycle construction. Sequences taken up to a circular shift of their components define cycles, the notation being C YC (B). Precisely, one has C YC (B) := S EQ (B)/S, where S is the equivalence relation between sequences defined by (α1 , . . , αr ) S (β1 , . . , βr ) iff there exists some circular shift τ of [1 .

Xk ) of integers (for some k) such that n = x1 + x2 + · · · + xk , xj ≥ 1. A partition of an integer n is a sequence (x1 , x2 , . . , xk ) of integers (for some k) such that n = x1 + x2 + · · · + xk and x1 ≥ x2 ≥ · · · ≥ xk . In both cases, the xi ’s are called the summands or the parts and the quantity n is called the size of the composition or the partition. By representing summands in unary using small discs (“•”), we can render graphically a composition by drawing bars between some of the balls; if we arrange summands vertically, compositions appear as ragged-landscapes.

1 − z n ) = n≥1 k∈Z It is proved formally and combinatorially in [76, p. 105]. As a consequence, the numbers √ {Pj }N j=0 can be determined in O(N N ) arithmetic operations. 18. A digital surprise. Define the constant 9 99 999 9999 ··· . ϕ := 10 100 1000 10000 Is it a surprise that it evaluates numerically to . 8900100999989990000001000099999999899999000000000010 · · · , that is, its decimal representation involves only the digits 0, 1, 8, 9? [This is suggested by a note of S. Ramanujan, “Some definite integrals”, Messenger of Math.

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