On two generalizations of directed lattice animals

Using two different generalizations of the normally ordered form of words in a locally-free group, we show that the first generalization relates the weighted directed lattice animals (DA) to the stationary configurations of the asymmetric simple exclusion process (ASEP), while the second generalization connects DA to the statistics of Dyck paths with fixed area, q-Airy functions and solutions of rational Painleve II equation. Based on joint works with N.Haug (first part) and with A.Gorsky and K. Bulycheva (second part).