The affine Stanley–Wilf growth-rate conjecture

Let $S_n(\tau)$ be the set of permutations in $S_n$ avoiding a pattern $\tau$, and let $\widetilde{S}^{/\!/}_n(\tau)$ be the set of bounded affine permutations of size $n$ avoiding $\tau$. A pattern $\tau$ is sum-indecomposable if it cannot be written as a nontrivial direct sum of permutations. Define the Stanley–Wilf limit by

$$L(\tau):=\lim_{n\rightarrow\infty}|S_n(\tau)|^{1/n}.$$

Affine Stanley–Wilf growth-rate conjecture. The proper growth rate

$$\lim_{n\rightarrow\infty}|\widetilde{S}^{/\!/}_n(\tau)|^{1/n}$$

exists and equals $L(\tau)$ for every sum-indecomposable pattern $\tau$. This conjecture predicts that imposing pattern avoidance on bounded affine permutations has the same exponential growth rate as imposing it on ordinary permutations. The paper establishes the corresponding lower bound via the injection from $S_n(\tau)$ into $\widetilde{S}^{/\!/}_n(\tau)$; the existence and equality of the affine growth rate are left open here.