  We study Hoeffding decomposable exchangeable sequences with values in a
finite set D. We provide a new combinatorial characterization of Hoeffding
decomposability and use this result to show that, if the cardinality of D is
strictly greater than 2, then there exists a class of neither P\'{o}lya nor
i.i.d. D-valued exchangeable sequences that are Hoeffding decomposable. The
construction of such sequences is based on some ideas appearing in Hill, Lane
and Sudderth [1987] and answers a question left open in El-Dakkak and Peccati
[2008].
