Abstract: Recent work on permutation equivariant neural networkshas mostly focused on the first order case (sets) and second order case (graphs). We describe the machinery for generalizing permutation equivariance to arbitrary $k$-ary interactions between entities for any value of $k$.We demonstrate the effectiveness of higher orderpermutation equivariant models on several real world applications and find that our resultscompare favorably to existing permutation invariant/equivariant baselines.