Discrete Mathematics & Theoretical Computer Science 
If F(x) = e^G(x), where F(x) = \Sum f(n)x^n and G(x) = \Sum g(n)x^n, with 0 ≤ g(n) = O(n^θn/n!), θ ∈ (0,1), and gcd(n : g(n) > 0) = 1, then f(n) = o(f(n − 1)). This gives an answer to Compton's request in Question 8.3 [Compton 1987] for an "easily verifiable sufficient condition" to show that an adequate class of structures has a labelled firstorder 01 law, namely it sufﬁces to show that the labelled component count function is O(n^θn) for some θ ∈ (0,1). It also provides the means to recursively construct an adequate class of structures with a labelled 01 law but not an unlabelled 01 law, answering Compton's Question 8.4.
Source : ScholeXplorer
IsRelatedTo ARXIV math/0511381 Source : ScholeXplorer IsRelatedTo DOI 10.1051/ps/2012007 Source : ScholeXplorer IsRelatedTo DOI 10.48550/arxiv.math/0511381
