  The $k$-dimensional functional order property ($\text{FOP}_k$) is a
combinatorial property of a $(k+1)$-partitioned formula. This notion arose in
work of Terry and Wolf, which identified $\text{NFOP}_2$ as a ternary analogue
of stability in the context of two finitary combinatorial problems related to
hypergraph regularity and arithmetic regularity. In this paper we show
$\text{NFOP}_k$ has equally strong implications in model-theoretic
classification theory, where its behavior as a $(k+1)$-ary version of stability
is in close analogy to the behavior of $k$-dependence as a $(k+1)$-ary version
of $\text{NIP}$. Our results include several new characterizations of
$\text{NFOP}_k$, including a characterization in terms of collapsing
indiscernibles, combinatorial recharacterizations, and a characterization in
terms of type-counting when $k=2$. As a corollary of our collapsing theorem, we
show $\text{NFOP}_k$ is closed under Boolean combinations, and that
$\text{FOP}_k$ can always be witnessed by a formula where all but one variable
have length $1$. When $k=2$, we prove a composition lemma analogous to that of
Chernikov and Hempel from the setting of $2$-dependence. Using this, we provide
a new class of algebraic examples of $\text{NFOP}_2$ theories. Specifically, we
show that if $T$ is the theory of an infinite dimensional vector space over a
field $K$, equipped with a bilinear form satisfying certain properties, then
$T$ is $\text{NFOP}_2$ if and only if $K$ is stable. Along the way we provide a
corrected and reorganized proof of Granger's quantifier elimination and
completeness results for these theories.
