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Abstract Thek-dimensional functional order property ($$\operatorname {FOP}_k$$ ${FOP}_k$) is a combinatorial property of a$$(k+1)$$ $(k+1)$-partitioned formula. This notion arose in work of Terry and Wolf [59, 60], which identified$$\operatorname {NFOP}_2$$ ${NFOP}_2$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$$\operatorname {NFOP}_k$$ ${NFOP}_k$has equally strong implications in model-theoretic classification theory, where its behavior as a$$(k+1)$$ $(k+1)$-ary version of stability is in close analogy to the behavior ofk-dependence as a$$(k+1)$$ $(k+1)$-ary version of$$\operatorname {NIP}$$ $NIP$. Our results include several new characterizations of$$\operatorname {NFOP}_k$$ ${NFOP}_k$, including a characterization in terms of collapsing indiscernibles, combinatorial recharacterizations, and a characterization in terms of type-counting when$$k=2$$ $k=2$. As a corollary of our collapsing theorem, we show$$\operatorname {NFOP}_k$$ ${NFOP}_k$is closed under Boolean combinations, and that$$\operatorname {FOP}_k$$ ${FOP}_k$can always be witnessed by a formula where all but one variable have length 1. When$$k=2$$ $k=2$, 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$$\operatorname {NFOP}_2$$ ${NFOP}_2$theories. Specifically, we show that ifTis the theory of an infinite dimensional vector space over a fieldK, equipped with a bilinear form satisfying certain properties, thenTis$$\operatorname {NFOP}_2$$ ${NFOP}_2$if and only ifKis stable. Along the way we provide a corrected and reorganized proof of Granger’s quantifier elimination and completeness results for these theories.