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Friday, January 21, 2011 |
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Maximum Stirling Numbers of the Second Kind |
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Say an integer $n$ is exceptional if the maximum Stirling number of the second kind $S(n,k)$ occurs for two(of necessity consecutive) values of $k$. We prove that the number of exceptional integers less than or equal to $x$ is $O(1)$. When $S(n,k)$ is the number of Stirling numbers of the first kind and an exceptional integer is defined analogously then Erdos proved that the only exceptional integer is is $n = 2$, no doubt this is true for the Stirling numbers of the second kind also. In spite of efforts by several authors this has still not been proved. The history of this problem will be surveyed. Perhaps the most interesting about this problem is the connection it demonstrates between calculus and combinatorics. In his book 'Mathematics and Its History' Stillwell points this out. He says Newton's calculus was based on formal power series, or as we say in our department generating functions. This is interesting to me because I never thought of it even though I inflicted many papers on any who read them on the topic treated in P. Flajolet and Stembridge's 'Analytic Combinatorics'. |