Svjetlan Feretić

Diagonally convex directed polyominoes and even trees: a bijection and related issues

We present a simple bijection between diagonally convex directed (DCD) polyominoes with n diagonals and plane trees with 2n edges in which every vertex has even degree (even trees), which specializes to a bijection between parallelogram polyominoes and full binary trees. Next we consider a natural definition of symmetry for DCD-polyominoes, even trees, ternary trees, and non-crossing trees, and show that the number of symmetric objects of a given size is the same in all four cases.

A new way of counting the column-convex polyominoes by perimeter

We introduce a new class of plane figures: the sequences of tailed column-convex polyominoes (for short: stapoes). Let G(x, y) and I(x, y) denote the perimeter generating functions for column-convex polyominoes and stapoes, respectively. It will be clear from the definitions that G(x, y) is a simple fraction of I(x, y). But this latter function can be DSV-computed by solving just one quadratic equation (and not a system of quadratic equations). Thus the formula for G(x, y) can be obtained with ease.

Combinatorics of diagonally convex directed polyominoes

Abstract A new bijection between the diagonally convex directed (dcd-) polyominoes and ternary trees makes it possible to enumerate the dcd-polyominoes according to several parameters (sources, diagonals, horizontal and vertical edges, target cells). For a part of these results we also give another proof, which is based on Raneys generalized lemma. Thanks to the fact that the diagonals of a dcd-polyomino can grow at most by one, the problem of q -enumeration of this object can be solved by an application of Gessels q -analog of the Lagrange inversion formula.

Polyominoes with nearly convex columns: A semi-directed model

Column-convex polyominoes are by now a well-explored model. So far, however, no attention has been given to polyominoes whose columns can have either one or two connected components. This little known kind of polyominoes seems not to be manageable as a whole. To obtain solvable models, one needs to introduce some restrictions. This paper is focused on polyominoes with hexagonal cells. The restrictions just mentioned are semi-directedness and an upper bound (say m ) on the size of the gap within a column. As the upper bound m grows, the solution of the model tends to break into more and more cases. We computed the area generating functions for m = 1, m = 2 and m = 3. In this paper, the m = 1 and m = 2 models are solved in full detail. To keep the size of the paper within reasonable limits, the result for the m = 3 model is stated without proof. The m = 1, m = 2 and m = 3 models have rational area generating functions, as column-convex polyominoes do. (It is practically sure, although we leave it unproved, that the area generating functions are also rational for m = 4, m = 5, ...) However, the growth constants of the new models are 4.114908 and more, whereas the growth constant of column-convex polyominoes is 3.863131.

A q -enumeration of convex polyominoes by the festoon approach

In 1938, Polya stated an identity involving the perimeter and area generating function for parallelogram polyominoes. To obtain that identity, Polya presumably considered festoons. A festoon (so named by Flajolet) is a closed path w which can be written as w = uv, where each step of u is either (1, 0) or (0, 1), and each step of v is either (-1, 0) or (0, -1).In this paper, we introduce four new festoon-like objects. As a result, we obtain explicit expressions (and not just identities) for the generating functions of parallelogram polyominoes, directed convex polyominoes, and convex polyominoes.

An alternative method for q-counting directed column-convex polyominoes

Abstract The area+perimeter generating function of directed column-convex polyominoes will be written as a quotient of two expressions, each of which involves powers of q of all kinds: positive, zero and negative. The method used in the proof applies to some other classes of column-convex polyominoes as well. At least occasionally, that method can do the case q =1 too.

Two generalizations of column-convex polygons

Column-convex polygons were first counted by area several decades ago, and the result was found to be a simple, rational, generating function. In this work we generalize that result. Let a p-column polyomino be a polyomino whose columns can have 1, 2, ..., p connected components. Then column-convex polygons are equivalent to 1-convex polyominoes. The area generating function of even the simplest generalization, namely 2-column polyominoes, is unlikely to be solvable. We therefore define two classes of polyominoes which interpolate between column-convex polygons and 2-column polyominoes. We derive the area generating functions of those two classes, using extensions of existing algorithms. The growth constants of both classes are greater than the growth constant of column-convex polyominoes. Rather tight lower bounds on the growth constants complement a comprehensive asymptotic analysis.

A q-enumeration of directed diagonally convex polyominoes

We enumerate directed diagonally convex polyominoes according to area. Our approach partly goes column by column, and partly goes row by row. In the end, we obtain fairly nice formulas.

A bijective perimeter enumeration of directed convex polyominoes

Abstract We give new bijective proofs for the following facts: (1) there are ( p+q p )( p+q q ) directed convex polyominoes with horizontal perimeter 2p+2 and vertical perimeter 2q+2 ; (2) there are ( 2n n ) directed convex polyominoes with total perimeter 2n+4 ; (3) the generating function Dc=∑ p,q=0 ∞ ( p+q p )( p+q q )x 2p+2 y 2q+2 is the square root of a rational function. Once (3) is proved, we have—and take—the opportunity to do an easy computation and obtain the known formula Dc= x 2 y 2 1−2x 2 −2y 2 +(x 2 −y 2 ) 2 .