Exchangeable Hoeffding decompositions over finite sets: A combinatorial characterization and counterexamples

We study Hoeffding decomposable exchangeable sequences with values in a finite set D={d1,…,dK}D={d1,…,dK}. We provide a new combinatorial characterization of Hoeffding decomposability and use this result to show that, for every K≥3K≥3, there exists a class of neither Pólya nor i.i.d. DD-valued exchangeable sequences that are Hoeffding decomposable.