Bijections between directed animals, multisets and Grand-Dyck paths (Q2181997)

Summary: An \(n\)-multiset of \([k]=\{1,2,\dots, k\}\) 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.