\(Q\)-grammars and wall polyominoes
A wall polyomino (or bar-graph polygon or solid-on-solid walk) is defined as a vertically convex polyomino in which all columns have their lowest cell at the same level. The upper path of such a polyomino can be bijectively coded with words using letters \(x\), \(y\), \(\overline y\) (corresponding to east, north and south steps). The language of nonempty words coding such polyominoes is algebraic and is generated by an unambiguous grammar. A slightly different grammar corresponds to coding directed positive words. The grammars lead to functional equations for generating functions for the polyominoes. The form of these equations, and some simple singularity computations, are used to prove that the area of wall polyominoes of perimeter \(2n\) has the Airy distribution as a limit law.
scientific article; zbMATH DE number 1347715