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We prove that for any fixed $n$, and for most permutation patterns $q$, thenumber $\textup{Av}_{n,\ell}(q)$ of $q$-avoiding permutations of length $n$that consist of $\ell$ skew blocks is a monotone decreasing function of $\ell$.We then show that this implies that for most patterns $q$, the generatingfunction $\sum_{n\geq 0} \textup{Av}_n(q)z^n$ of the sequence$\textup{Av}_n(q)$ of the numbers of $q$-avoiding permutations is not rational.Placing our results in a broader context, we show that for rational powerseries $F(z)$ and $G(z)$ with nonnegative real coefficients, the relation$F(z)=1/(1-G(z))$ is supercritical, while for most permutation patterns $q$,the corresponding relation is not supercritical.

Most principal permutation classes have nonrational generating functions