Two associative operads of packed words

The associative operad is a central structure in operad theory, defined on the linear span of the set of permutations. We build two analogs of the associative operad on the linear span of the set of packed words which turn out to be set-theoretical. By seeing a packed word as a surjective map between two finite sets, our first operad is graded by the cardinality of the domain and the second one, by the cardinality of the codomain. In the same way as the associative operad of permutations contains as quotients the duplicial and interstice operads, we derive similar structures for our operads of packed words. We propose also an analogue of Dynkin idempotent of Zie algebras in this context of operads of packed words.