Asymptotic estimates of Stirling numbers
01 Jan 1992-pp 1-9
About: The article was published on 1992-01-01 and is currently open access. It has received 85 citations till now. The article focuses on the topics: Stirling number & Asymptotic analysis.
Citations
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01 Mar 1996
TL;DR: In this article, the saddle point and large singularities of analytic functions are used for graphical enumeration, including implicit functions, recurrences, and combinations of methods, which are related to our work.
Abstract: 12 Large singularities of analytic functions 113 12.1 The saddle point 13 Multivariate generating functions 128 14 Mellin and other integral transforms 134 15 Functional equations, recurrences, and combinations of methods 137 15.1 Implicit functions, graphical enumeration, and related
385 citations
TL;DR: In this article, it was shown that various well-known generalizations of Stirling numbers of the first and second kinds can be unified by starting with transformations between generalized factorials involving three arbitrary parameters.
156 citations
TL;DR: Alignments and approximations for finite sequences, called Polya frequency sequences, which follow from their probabilistic representation are reviewed, finding a number of improvements of known estimates.
142 citations
16 Nov 2002
TL;DR: In this paper, the existence of the 3-XORSAT threshold was proved and the value of the threshold was established as a function of the root of a transcendental equation.
Abstract: We prove the existence of the 3-XORSAT threshold, establishing its value as a function of the root of a transcendental equation.
130 citations
TL;DR: A general upper bound of the threshold forr-SAT formulae is established giving a value for 3-S AT of 4.64, significantly improving the previous best upper bound.
113 citations
References
More filters
Book•
01 Mar 1996
TL;DR: In this article, the saddle point and large singularities of analytic functions are used for graphical enumeration, including implicit functions, recurrences, and combinations of methods, which are related to our work.
Abstract: 12 Large singularities of analytic functions 113 12.1 The saddle point 13 Multivariate generating functions 128 14 Mellin and other integral transforms 134 15 Functional equations, recurrences, and combinations of methods 137 15.1 Implicit functions, graphical enumeration, and related
385 citations
TL;DR: In this article, it was shown that various well-known generalizations of Stirling numbers of the first and second kinds can be unified by starting with transformations between generalized factorials involving three arbitrary parameters.
156 citations
TL;DR: Alignments and approximations for finite sequences, called Polya frequency sequences, which follow from their probabilistic representation are reviewed, finding a number of improvements of known estimates.
142 citations
TL;DR: A general upper bound of the threshold forr-SAT formulae is established giving a value for 3-S AT of 4.64, significantly improving the previous best upper bound.
113 citations
27 Mar 2009
TL;DR: This paper proposes a determinisation technique that is simpler than the constructions of Safra, Piterman, and Muller and Schupp, because it separates the principle acceptance mechanism from the concrete acceptance condition.
Abstract: The introduction of an efficient determinisation technique for Buchi automata by Safra has been a milestone in automata theory. To name only a few applications, efficient determinisation techniques for *** -word automata are the basis for several manipulations of *** -tree automata (most prominently the nondeterminisation of alternating tree automata) as well as for satisfiability checking and model synthesis for branching- and alternating-time logics. This paper proposes a determinisation technique that is simpler than the constructions of Safra, Piterman, and Muller and Schupp, because it separates the principle acceptance mechanism from the concrete acceptance condition. The principle mechanism intuitively uses a Rabin condition on the transitions; we show how to obtain an equivalent Rabin transition automaton with approximately (1.65 n ) n states from a nondeterministic Buchi automaton with n states. Having established this mechanism, it is simple to develop translations to automata with standard acceptance conditions. We can construct standard Rabin automata whose state-space is bilinear in the size of the input alphabet and the state-space of the Rabin transition automaton, or, for large input alphabets, contains approximately (2.66 n ) n states, respectively. We also provide a flexible translation to parity automata with O (n !2) states and 2n priorities based on a later introduction record, and hence connect the transformation of the acceptance condition to other record based transformations known from the literature.
105 citations