Equivalence of the two-dimensional directed animal problem to a one- dimensional path problem.

*(English)*Zbl 0727.05036Summary: We introduce a one-to-one correspondence between directed animals on a square lattice and a class of one-dimensional paths. We derive very simply the formulae giving the exact number of directed animals of given size and the average width of such animals. The more surprising result is the fact that the number of compact-rooted directed animals of size n is \(3^{n-1}\).

##### MSC:

05C38 | Paths and cycles |

82B20 | Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics |

##### Keywords:

directed animals
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\textit{D. Gouyou-Beauchamps} and \textit{G. Viennot}, Adv. Appl. Math. 9, No. 3, 334--357 (1988; Zbl 0727.05036)

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