| We demonstrate asymptotic properties of some popular combinatorial objects, including a partial answer to an open problem posed by Michael Atkinson and a general result on conditions for the coincidence of asymptotic normality and uniformity.;For a permutation pi ∈ Sn written in one line notation as pi = pi1pi2...pi n, we say pi contains the pattern sigma ∈ Sm if there exists a subsequence pi1=pim such that for all 1 ≤ j; k ≤ m it holds that pij<pik if and only if sigmaj < sigma k. Denote by sn(tau,sigma) the number of n-permutations which do not contain either of the patterns sigma and tau. Letting sigma' denote the pattern (m + 1)sigma; we construct classes of patterns for which the limit supremum of sn(123... r, sigma)1/n agrees with the limit supremum of sn(123...r, sigma')1/n for several classes of patterns sigma. We also construct classes of permutations which avoid 123...r and contain many patterns.;Many combinatorial sequences are of the form (an,k) where n ranges over the non-negative integers N and, for each n, there exists m = m( n) such that k ranges from 1 to m. We call such a sequence a combinatorial distribution. Many combinatorial distributions, upon rescaling, approach in distribution the normal distribution as n grows to infinity, a phenomenon we call asymptotic normality. A combinatorial distribution is said to be asymptotically uniform if, for each positive integer q and each residue class modulo q, the sum of coefficients an,k with k ≡ r (mod q) approaches 1/q as n grows to infinity. We call this asymptotic uniformity. We prove that if the generating polynomials for a combinatorial distribution have real, nonnegative zeros, asymptotic normality implies asymptotic uniformity. We apply this result to several sequences from the literature.;Finally, we present original results on the zeros of the Bell polynomials which were rst attained in proving the asymptotic uniformity of the Stirling numbers of the second kind Sn,k. |
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