Polyominoes with nearly convex columns: A model with semidirected blocks

In most of today\u27s exactly solved classes of polyominoes, either all members are convex (in some way), or all members are directed, or both. If the class is neither convex nor directed, the exact solution is usually elusive. This paper is focused on polyominoes with hexagonal cells. Concretely, we deal with polyominoes whose columns can have either one or two connected components. Those polyominoes (unlike the well-explored column-convex polyominoes) cannot be exactly enumerated by any of the now existing methods. It is therefore appropriate to introduce additional restrictions, thus obtaining solvable subclasses. In our recent paper, published in this same journal, the restrictions just mentioned were semidirectedness and an upper bound on the size of the gap within a column. In this paper, the semidirectedness requirement is made looser. The result is that now the exactly solved subclasses are larger and have greater growth constants. These new polyomino families also have the advantage of being invariant under the reflection about the vertical axis

The Area Generating Function for the Column-Convex Polyominoes on the Checkerboard Lattice

The aim of the present work is to compute the area generating function (gf) for the column-convexpolyominoes on the checkerboard lattice. It is interesting that this area gf includes as two special cases the area gfs for the rectangular and honeycomb lattices. The problem treated here is complementary to the problem concerning the perimeter gfs, which was suggested by Wu and solved by Tzeng and Lin

Combinatorics of diagonally convex directed polyominoes

AbstractA 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 Raney's 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 Gessel's q-analog of the Lagrange inversion formula

A New Coding for Column-Convex Directed Animals

The present article contains two new results: a closed form expression for the number of column-convex directed (ccd-) animals having a given bond perimeter, directed site perimeter and number of columns, as well as a certain logarithmic function, a part of which is the ccd-animals two perimeters & columns generating function. Finally, an attempt has been made to formulate, in a more immediate way, the original proof of Delest and Dulucq1 concerning the number of ccd-animals with a given area

The Column-Convex Polyominoes Perimeter Generating Function for Everybody

The function mentioned in the title will be leisurely derived in two different ways. The apparatus used in the proofs consists of easy bijections and some generating functionology

The Area Generating Function for the Column-Convex Polyominoes on the Checkerboard Lattice

The aim of the present work is to compute the area generating function (gf) for the column-convexpolyominoes on the checkerboard lattice. It is interesting that this area gf includes as two special cases the area gfs for the rectangular and honeycomb lattices. The problem treated here is complementary to the problem concerning the perimeter gfs, which was suggested by Wu and solved by Tzeng and Lin

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

AbstractThe 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

A q-enumeration of convex polyominoes by the festoon approach

AbstractIn 1938, Pólya stated an identity involving the perimeter and area generating function for parallelogram polyominoes. To obtain that identity, Pólya 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

A perimeter enumeration of column-convex polyominoes

CombinatoricsThis work is concerned with the perimeter enumeration of column-convex polyominoes. We consider both the rectangular lattice and the hexagonal lattice. For the rectangular lattice, two formulas for the generating function (gf) already exist and, to all appearances, neither of them admits of a further simplification. We first rederive those two formulas (so as to make the paper self-contained), and then we enrich the rectangular lattice gf with some additional variables. That done, we make a change of variables, which (practically) produces the hexagonal lattice gf. This latter gf was first found by Lin and Wu in 1990. However, our present formula, in addition to having a simpler form, also allows a substantially easier Taylor series expansion. As to the methods, our one is descended from algebraic languages, whereas Lin and Wu used the Temperley methodology