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Extra resources for Asymptotic enumeration methods
It is based on Chebyshev’s inequality, which says 49 that if X is a real-valued random variable with finite second moment E(X 2 ), then Prob (|X − E(X)| ≥ α|E(X)|) ≤ E(X 2 ) − E(X)2 . 86) that is often used is Prob (X = 0) ≤ E(X 2 ) − E(X)2 . 87) are usually applied for X = Y 1 + · · · + Yn , where the Yj are other random variables. The helpful feature of the inequalities is that they require only knowledge of the pairwise dependencies among the Y j , which is easier to study than the full joint distribution of the Y j .
1. For example, they can be used to prove the results of  on volumes of spheres in the Lee metric. 1 can be generalized in a straightforward way to multivariate generating functions. 43) converges we have am,n ≤ x−m y −n f (x, y) . 1 to multivariate functions can also be derived, but are again harder than the upper bound . 2. 1.
Let t n be the number of rooted labeled trees on n vertices, so that t1 = 1, t2 = 2, t3 = 9. ) Let ∞ tn t(z) = n=1 zn n! 17) be the exponential generating function. If we remove the root of a rooted labeled tree with n vertices, we are left with k ≥ 0 rooted labeled trees that contain a total of n − 1 vertices. The total number of ways of arranging an ordered selection of k rooted trees with a total of n − 1 vertices is [z n−1 ]t(z)k . 35 Since the order of the trees does not matter, we have 1 n−1 [z ]t(z)k k!