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Abstract:
A permutation $\pi=(\pi_{1},\ldots,\pi_{n})$ is consecutive
$123$-avoiding if there is no sequence of the form $\pi_{i} <
\pi_{i+1} < \pi_{i+2}$. More generally, for $S$ a collection of
permutations on $m+1$ elements, this definition extends to define
consecutive $S$-avoiding permutations. We show that the spectrum of
an associated integral operator on the space $L^{2}[0,1]^{m}$
determines the asymptotics of the number of consecutive $S$-avoiding
permutations. Moreover, using an operator version of the classical
Frobenius-Perron theorem due to Kre\u{\i}n and Rutman, we prove
asymptotic results for large classes of patterns $S$. This extends
previously known results of Elizalde. This is joint work with Sergey
Kitaev and Peter Perry.
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