Bijections between Directed Animals, Multisets and Grand-Dyck Paths

  • Jean-Luc Baril
  • David Bevan
  • Sergey Kirgizov


An $n$-multiset of $[k]=\{1,2,\ldots, k\}$ 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.

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