# Bijections between directed animals, multisets and Grand-Dyck paths

## TL;DR

This paper establishes bijections between directed animals, multisets with no consecutive elements, and Grand-Dyck paths, providing new combinatorial insights and enumerations for these structures.

## Contribution

It introduces constructive bijections connecting directed animals, multisets, and Grand-Dyck paths, along with their statistics, and derives the bivariate generating function for multisets without consecutive elements.

## Key findings

- Derived the bivariate generating function for multisets with no consecutive elements.
- Established bijections between directed animals, multisets, and Grand-Dyck paths.
- Transported classical and new statistics through the bijections.

## Abstract

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.