In this paper we consider a restricted class of convex polyominoes that we call Z-convex polyominoes. Z-convex polyominoes are polyominoes such that any two pairs of cells can be connected by a monotone path making at most two turns (like the letter Z). In particular they are convex polyominoes, but they appear to resist standard decompositions. We propose a construction by ``inflation'' that allows to write a system of functional equations for their generating functions. The generating function P(t) of Z-convex polyominoes with respect to the semi-perimeter turns out to be algebraic all the same and surprisingly, like the generating function of convex polyominoes, it can be expressed as a rational function of t and the generating function of Catalan numbers.
We present a new method to obtain the generating functions for directed convex polyominoes according to several different statistics including: width, height, size of last column/row and number of corners. This method can be used to study different families of directed convex polyominoes: symmetric polyominoes, parallelogram polyominoes. In this paper, we apply our method to determine the generating function for directed k-convex polyominoes. We show it is a rational function and we study its asymptotic behavior.
In this paper, we introduce and study {\it Carlitz polyominoes}. In particular, we show that, as $n$ grows to infinity, asymptotically the number of \begin{enumerate} \item column-convex Carlitz polyominoes with perimeter $2n$ is \beq \frac{9\sqrt{2}(14+3\sqrt{3})}{2704\sqrt{πn^3}}4^n. \feq \item convex Carlitz polyominoes with perimeter $2n$ is \beq \frac{n+1}{10}\left(\frac{3+\sqrt{5}}{2}\right)^{n-2}. \feq \end{enumerate}
Column-convex polyominoes were introduced in 1950's by Temperley, a mathematical physicist working on "lattice gases". By now, column-convex polyominoes are a popular and well-understood model. There exist several generalizations of column-convex polyominoes; an example is a model called multi-directed animals. In this paper, we introduce a new sequence of supersets of column-convex polyominoes. Our model (we call it level m column-subconvex polyominoes) is defined in a simple way. We focus on the case when cells are hexagons and we compute the area generating functions for the levels one and …
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 semidirectedness and an upper bound on the size of the gap within a column. The solvable models so obtained have rational area generating functions, as column-convex polyominoe…
We give a new combinatorial proof for the number of convex polyominoes whose minimum enclosing rectangle has given dimensions. We also count the subclass of these polyominoes that contain the lower left corner of the enclosing rectangle (directed polyominoes). We indicate how to sample random polyominoes in these classes. As a side result, we calculate the first and second moments of the number of common points of two monotone lattice paths between two given points.
In this article we prove a conjecture of Bezdek, Brass, and Harborth concerning the maximum volume of the convex hull of any facet-to-facet connected system of n unit hypercubes in the d-dimensional Euclidean space. For d=2 we enumerate the extremal polyominoes and determine the set of possible areas of the convex hull for each n.
This paper concerns the enumeration of rotation-type and congruence-type convex polyominoes on the square lattice. These can be defined as orbits of the groups C4, of rotations, and D4, of symmetries of the square acting on (translation- type) polyominoes. In virtue of Burnside's Lemma, it is sufficient to enumerate the various symmetry classes (fixed points) of polyominoes defined by the elements of C4 and D4. Using the Temperley--Bousquet-Melou methodology, we solve this problem and provide explicit or recursive formulas for their generating functions according to width, height and area. We …
Lin and Chang gave a generating function of convex polyominoes with an $m+1$ by $n+1$ minimal bounding rectangle. Gessel showed that their result implies that the number of such polyominoes is $$ \frac{m+n+mn}{m+n}{2m+2n\choose 2m}-\frac{2mn}{m+n}{m+n\choose m}^2. $$ We show that this result can be derived from some binomial coefficients identities related to the generating function of Jacobi polynomials.
Parallelogram polyominoes are a subclass of convex polyominoes in the square lattice that has been studied extensively in the literature. Recently congruence classes of convex polyominoes with respect to rotations and reflections have been enumerated by counting orbits under the action of the dihedral group D4, of symmetries of the square, on (translation-type) convex polyominoes. Asymmetric convex polyominoes were also enumerated using Moebius inversion in the lattice of subgroups of D4. Here we extend these results to the subclass of parallelogram polyominos using a subgroup D2 of D4 which a…
Motivated by some binomial coefficients identities encountered in our approach to the enumeration of convex polyominoes, we prove some more general identities of the same type, one of which turns out to be related to a strange evaluation of ${}_3F_2$ of Gessel and Stanton.
In most of today's 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 uses to be 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…
The main contribution of this paper is a new column-by-column method for the decomposition of generating functions of convex polyominoes suitable for enumeration with respect to various statistics including but not limited to interior vertices, boundary vertices of certain degrees, and outer site perimeter. Using this decomposition, among other things, we show that A) the average number of interior vertices over all convex polyominoes of perimeter $2n$ is asymptotic to $\frac{n^2}{12}+\frac{n\sqrt{n}}{3\sqrtπ} -\frac{(21π-16)n}{12π}.$ B) the average number of boundary vertices with degree two …
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 chapter 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 to 2-column polyominoes, is unlikely to be solvable. We therefore define a class of polyominoes which interpolates between column-convex polygons and 2-column polyominoes. We d…
In this thesis, we consider the problem of characterizing and enumerating sets of polyominoes described in terms of some constraints, defined either by convexity or by pattern containment. We are interested in a well known subclass of convex polyominoes, the k-convex polyominoes for which the enumeration according to the semi-perimeter is known only for k=1,2. We obtain, from a recursive decomposition, the generating function of the class of k-convex parallelogram polyominoes, which turns out to be rational. Noting that this generating function can be expressed in terms of the Fibonacci polyno…
A polyomino is a finite, edge-connected set of cells in the plane. At the present time, an enumeration of all polyominoes is nowhere in sight. On the other hand, there are several subsets of polyominoes for which generating functions are known. For example, there exists extensive knowledge about column-convex polyominoes, a model introduced by Temperley in 1956. While studying column-convex polyominoes, researchers also gave a look at diagonally convex polyominoes (DCPs), but noticed an awkward feature: when the last diagonal of a DCP is deleted, the remaining object is not always a polyomino.…
We study the coordinate ring of an $L$-convex polyomino, determine its regularity in terms of the maximal number of rooks that can be placed in the polyomino. We also characterize the Gorenstein $L$-convex polyominoes and those which are Gorenstein on the punctured spectrum, and compute the Cohen--Macaulay type of any $L$-convex polyomino in terms of the maximal rectangles covering it.
Hexagonal polyominoes are polyominoes on the honeycomb lattice. We enumerate the symmetry classes of convex hexagonal polyominoes. Here convexity is to be understood as convexity along the three main column directions. We deduce the generating series of free (i.e. up to reflection and rotation) and of asymmetric convex hexagonal polyominoes, according to area and half-perimeter. We give explicit formulas or implicit functional equations for the generating series, which are convenient for computer algebra.
We classify all convex polyomino ideals which are linearly related or have a linear resolution. Convex stack polyominoes whose ideals are extremal Gorenstein are also classified. In addition, we characterize, in combinatorial terms, the distributive lattices whose join-meet ideals are extremal Gorenstein or have a linear resolution.
Back in the early days of polyomino enumeration, a model called column-convex polyominoes was introduced and its area generating function was found. That generating function is rational: the numerator has degree four and the denominator has degree three. Let a column-duplex polyomino be a polyomino whose columns can have either one or two connected components. A simplex-duplex polyomino is a column-duplex polyomino in which there is no occurrence of two adjacent columns each having two connected components. Simplex-duplex polyominoes are not easy to deal with, but their area generating functio…
As a generalization of polyominoes we consider edge-to-edge connected nonoverlapping unions of regular $k$-gons. For $n\le 4$ we determine formulas for the number $a_k(n)$ of generalized polyominoes consisting of $n$ regular $k$-gons. Additionally we give a table of the numbers $a_k(n)$ for small $k$ and $n$ obtained by computer enumeration. We finish with some open problems for $k$-polyominoes.
We study the statistics area, bounce and dinv on the set of parallelogram polyominoes having a rectangular m times n bounding box. We show that the bi-statistics (area, bounce) and (area, dinv) give rise to the same q,t-analogue of Narayana numbers which was introduced by two of the authors in [arXiv:1208.0024]. We prove the main conjectures of that paper: the q,t-Narayana polynomials are symmetric in both q and t, and m and n. This is accomplished by providing a symmetric functions interpretation of the q,t-Narayana polynomials which relates them to the famous diagonal harmonics.
We introduce a family of 3D combinatorial objects that we define as minimal 3D polyominoes inscribed in a rectanglar prism. These objects are connected sets of unitary cubic cells inscribed in a given rectangular prism and of minimal volume under this condition. They extend the concept of 2D polyominoes inscribed in a rectangle defined in a previous work. Using their geometric structure and elementary combinatorial arguments, we construct generating functions of minimal 3D polyominoes in the form of rational functions. We also obtain a number of exact formulas and recurrences for sub-families …
Let $P(n)$ be the number of polyominoes of $n$ cells and $λ$ be Klarner's constant, that is, $λ=\lim_{n\to\infty} \sqrt[n]{P(n)}$. We show that there exist some positive numbers $A,T$, so that for every $n$ \[ P(n) \ge An^{-T\log n} λ^n. \] This is somewhat a step toward the well known conjecture that there exist positive $C,θ$ so that $P(n)\sim Cn^{-θ}λ^n$ for every $n$. In fact, if we assume another popular conjecture that $P(n)/P(n-1)$ is increasing, we can get rid of $\log n$ to have \[ P(n)\ge An^{-T}λ^n. \] Beside the above theoretical result, we also conjecture that the ratio of the num…
We put recent results on the symmetry of the joint distribution of the numbers of crossings and nestings of two edges over matchings, set partitions and linked partitions, in the larger context of the enumeration of increasing and decreasing chains of length 2 in fillings of moon polyominoes.
We prove that the number of 01-fillings of a given stack polyomino (a polyomino with justified rows whose lengths form a unimodal sequence) with at most one 1 per column which do not contain a fixed-size northeast chain and a fixed-size southeast chain, depends only on the set of row lengths of the polyomino. The proof is via a bijection between fillings of stack polyominoes which differ only in the position of one row and uses the Hecke insertion algorithm by Buch, Kresch, Shimozono, Tamvakis, and Yong and the jeu de taquin for increasing tableaux of Thomas and Yong. Moreover, our bijection g…
Let $X$ be a convex polyomino such that its vertex set is a sublattice of $\mathbb{N}^2$. Let $\Bbbk[X]$ be the toric ring (over a field $\Bbbk$) associated to $X$ in the sense of Qureshi, \emph{J. Algebra}, 2012. Write the Hilbert series of $\Bbbk[X]$ as $(1 + h_1 t + h_2 t^2 + \cdots )/(1-t)^{\dim(\Bbbk[X])}$. For $k \in \mathbb{N}$, let $r_k$ be the number of configurations in $X$ with $k$ pairwise non-attacking rooks. We show that $h_2 < r_2$ if $X$ is not a thin polyomino. This partially confirms a conjectured characterization of thin polyominoes by Rinaldo and Romeo, \emph{J. Algebraic C…
We provide a new succession rule (i.e. generating tree) associated with Schröder numbers, that interpolates between the known succession rules for Catalan and Baxter numbers. We define Schröder and Baxter generalizations of parallelogram polyominoes, called slicings, which grow according to these succession rules. In passing, we also exhibit Schröder subclasses of Baxter classes, namely a Schröder subset of triples of non-intersecting lattice paths, a new Schröder subset of Baxter permutations, and a new Schröder subset of mosaic floorplans. Finally, we define two families of subclasses of Bax…
We introduce the $k$-bonacci polyominoes, a new family of polyominoes associated with the binary words avoiding $k$ consecutive $1$'s, also called generalized $k$-bonacci words. The polyominoes are very entrancing objects, considered in combinatorics and computer science. The study of polyominoes generates a rich source of combinatorial ideas. In this paper we study some properties of $k$-bonacci polyominoes. Specifically, we determine their recursive structure and, using this structure, we enumerate them according to their area, semiperimeter, and length of the corresponding words. We also in…
In this paper we determine a closed formula for the number of convex permutominoes of size n. We reach this goal by providing a recursive generation of all convex permutominoes of size n+1 from the objects of size n, according to the ECO method, and then translating this construction into a system of functional equations satisfied by the generating function of convex permutominoes. As a consequence we easily obtain also the enumeration of some classes of convex polyominoes, including stack and directed convex permutominoes.
We establish a stronger symmetry between the numbers of northeast and southeast chains in the context of 01-fillings of moon polyominoes. Let $\M$ be a moon polyomino with $n$ rows and $m$ columns. Consider all the 01-fillings of $\M$ in which every row has at most one 1. We introduce four mixed statistics with respect to a bipartition of rows or columns of $\M$. More precisely, let $S \subseteq \{1,2,..., n\}$ and $\mathcal{R}(S)$ be the union of rows whose indices are in $S$. For any filling $M$, the top-mixed (resp. bottom-mixed) statistic $α(S; M)$ (resp. $β(S; M)$) is the sum of the numbe…
What is the maximum number of holes that a polyomino with $n$ tiles can enclose? Call this number $f(n)$. We show that if $n_k = \left( 2^{2k+1} + 3 \cdot 2^{k+1}+4 \right) / 3$ and $h_k = \left( 2^{2k}-1 \right) /3$, then $f(n_k) = h_k$ for $k \ge 1$. We also give nearly matching upper and lower bounds for large $n$, showing as a corollary that $f(n) \approx n/2$.
We give a complete solution to the extremal topological combinatorial problem of finding the minimum number of tiles needed to construct a polyomino with $h$ holes. We denote this number by $g(h)$ and say that a polyomino is crystallized if it has $h$ holes and $g(h)$ tiles. We analyze structural properties of crystallized polyominoes and characterize their efficiency by a topological isoperimetric inequality that relates minimum perimeter, the area of the holes, and the structure of the dual graph of a polyomino. We also develop a new dynamical method of creating sequences of polyominoes whic…
We present recursive formulas giving the maximal number of leaves in tree-like polyforms living in two-dimensional regular lattices and in tree-like polycubes in the three-dimensional cubic lattice. We call these tree-like polyforms and polycubes \emph{fully leafed}. The proof relies on a combinatorial algorithm that enumerates rooted directed trees that we call abundant. In the last part, we concentrate on the particular case of polyforms and polycubes, that we call \emph{saturated}, which is the family of fully leafed structures that maximize the ratio $\mbox{(number of leaves)}/\mbox{ (numb…
Fighting fish were very recently introduced by the authors as combinatorial structures made of square tiles that form two dimensional branching surfaces. A main feature of these fighting fish is that the area of uniform random fish of size $n$ scales like $n^{5/4}$ as opposed to the typical $n^{3/2}$ area behavior of the staircase or direct convex polyominoes that they generalize. In this extended abstract we concentrate on enumerative properties of fighting fish: in particular we provide a new decomposition and we show that the number of fighting fish with $i$ left lower free edges and $j$ ri…
In this paper we study relations between connected and weakly convex domination numbers. We show that in general the difference between these numbers can be arbitrarily large and we focus on the graphs for which a weakly convex domination number equals a connected domination number. We also study the influence of the edge removing on the weakly convex domination number, in particular we prove that the weakly convex domination number is an interpolating function.
The q-Catalan numbers studied by Carlitz and Riordan are polynomials in q with nonnegative coefficients. They evaluate, at q=1, to the Catalan numbers: 1, 1, 2, 5, 14,..., a log-convex sequence. We use a combinatorial interpretation of these polynomials to prove a q-log-convexity result. The sequence of q-Catalan numbers is not q-log-convex in the narrow sense used by other authors, so our work suggests a more flexible definition of q-log convex be adopted.
In this paper, we obtain the counting formulaes of convex pentagons and convex hexagons, respectively, in an $n$-triangular net by solving the corresponding recursive formulaes.
A set of vertices $S$ of a graph $G$ is a (geodesic)convex set, if $S$ contains all the vertices belonging to any shortest path connecting between two vertices of $S$. The cardinality of maximum proper convex set of $G$ is called the convexity number, con$(G)$ of $G$. The complementary prism $G\bar{G}$ of $G$ is obtained from the disjoint union of $G$ and its complement $\bar{G}$ by adding the edges of a perfect matching between them. In this work, we examine the convex sets of the complementary prism of a tree and derive formulas for the convexity numbers of the complementary prisms of all tr…
