  For a permutation $\pi:[k] \to [k]$, a function $f:[n] \to \mathbb{R}$
contains a $\pi$-appearance if there exists $1 \leq i_1 < i_2 < \dots < i_k
\leq n$ such that for all $s,t \in [k]$, $f(i_s) < f(i_t)$ if and only if
$\pi(s) < \pi(t)$. The function is $\pi$-free if it has no $\pi$-appearances.
In this paper, we investigate the problem of testing whether an input function
$f$ is $\pi$-free or whether $f$ differs on at least $\varepsilon n$ values
from every $\pi$-free function. This is a generalization of the well-studied
monotonicity testing and was first studied by Newman, Rabinovich,
Rajendraprasad and Sohler (Random Structures and Algorithms 2019). We show that
for all constants $k \in \mathbb{N}$, $\varepsilon \in (0,1)$, and permutation
$\pi:[k] \to [k]$, there is a one-sided error $\varepsilon$-testing algorithm
for $\pi$-freeness of functions $f:[n] \to \mathbb{R}$ that makes
$\tilde{O}(n^{o(1)})$ queries. We improve significantly upon the previous best
upper bound $O(n^{1 - 1/(k-1)})$ by Ben-Eliezer and Canonne (SODA 2018). Our
algorithm is adaptive, while the earlier best upper bound is known to be tight
for nonadaptive algorithms.
