Bijections between directed animals, multisets and Grand-Dyck paths

Baril, Jean-Luc and Bevan, David and Kirgizov, Sergey (2020) Bijections between directed animals, multisets and Grand-Dyck paths. The Electronic Journal of Combinatorics, 27 (2). P2.10. ISSN 1077-8926 (

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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.