Bijections between directed animals, multisets and Grand-Dyck paths

An n-multiset of [k] = {1, 2,..., k} consists of a set of n elements from [k] where each element can be repeated. We present the bivariate generating function for n-multisets of [k] with no consecutive elements. For n = k, these multisets have the same enumeration as directed animals in the square lattice. Then we give constructive bijections between directed animals, multisets with no consecutive elements and Grand-Dyck paths avoiding the pattern DUD, and we show how classical and novel statistics are transported by these bijections.