  Permutation pattern-avoidance is a central concept of both enumerative and
extremal combinatorics. In this paper we study the effect of permutation
pattern-avoidance on the complexity of optimization problems.
  In the context of the dynamic optimality conjecture (Sleator, Tarjan, STOC
1983), Chalermsook, Goswami, Kozma, Mehlhorn, and Saranurak (FOCS 2015)
conjectured that the amortized search cost of an optimal binary search tree
(BST) is constant whenever the search sequence is pattern-avoiding. The best
known bound to date is $2^{\alpha{(n)}(1+o(1))}$ recently obtained by
Chalermsook, Pettie, and Yingchareonthawornchai (SODA 2024); here $n$ is the
BST size and $\alpha(\cdot)$ the inverse-Ackermann function. In this paper we
resolve the conjecture, showing a tight $O(1)$ bound. This indicates a barrier
to dynamic optimality: any candidate online BST (e.g., splay trees or greedy
trees) must match this optimum, but current analysis techniques only give
superconstant bounds.
  More broadly, we argue that the easiness of pattern-avoiding input is a
general phenomenon, not limited to BSTs or even to data structures. To
illustrate this, we show that when the input avoids an arbitrary, fixed, a
priori unknown pattern, one can efficiently compute a $k$-server solution of
$n$ requests from a unit interval, with total cost $n^{O(1/\log k)}$, in
contrast to the worst-case $\Theta(n/k)$ bound; and a traveling salesman tour
of $n$ points from a unit box, of length $O(\log{n})$, in contrast to the
worst-case $\Theta(\sqrt{n})$ bound; similar results hold for the euclidean
minimum spanning tree, Steiner tree, and nearest-neighbor graphs.
  We show both results to be tight. Our techniques build on the Marcus-Tardos
proof of the Stanley-Wilf conjecture, and on the recently emerging concept of
twin-width.
