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{\bf Edward A. Bender, William J. Helton and L. Bruce Richmond}
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{\bf Asymptotics of Permutations with
Nearly Periodic Patterns of Rises and Falls}
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Ehrenborg obtained asymptotic results for nearly alternating
permutations and conjectured an asymptotic formula for the number of
permutations that have a nearly periodic run pattern.  We prove a
generalization of this conjecture, rederive the fact that the
asymptotic number of permutations with a periodic run pattern has the
form $Cr^{-n}\,n!$, and show how to compute the various constants.  A
reformulation in terms of iid random variables leads to an eigenvalue
problem for a Fredholm integral equation.  Tools from functional
analysis establish the necessary properties.

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