[Paper Review] Enumeration of polyominoes defined in terms of pattern avoidance or convexity constraints

Daniela Battaglino|arXiv (Cornell University)|May 13, 2014

Advanced Combinatorial Mathematics19 references3 citations

TL;DR

This dissertation introduces a novel framework for enumerating polyominoes using pattern avoidance and convexity constraints, linking them to permutation classes via submatrix avoidance in binary matrices. It establishes connections between polyomino classes and permutation patterns, derives generating functions for $k$-parallelogram polyominoes, and provides a bijective proof linking them to planted plane trees, offering new combinatorial insights and enumeration techniques for structured polyomino families.

ABSTRACT

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 polynomials, we describe a bijection between the class of k-parallelogram polyominoes and the class of planted planar trees having height less than k+3. In the second part of the thesis we examine the notion of pattern avoidance, which has been extensively studied for permutations. We introduce the concept of pattern avoidance in the context of matrices, more precisely permutation matrices and polyomino matrices. We present definitions analogous to those given for permutations and in particular we define polyomino classes, i.e. sets downward closed with respect to the containment relation. So, the study of the old and new properties of the redefined sets of objects has not only become interesting, but it has also suggested the study of the associated poset. In both approaches our results can be used to treat open problems related to polyominoes as well as other combinatorial objects.

Motivation & Objective

• To develop a systematic approach to classifying and enumerating polyominoes based on convexity and pattern avoidance constraints.
• To establish a correspondence between polyomino classes and permutation classes using submatrix avoidance in binary matrices.
• To explore the structural and enumerative properties of $k$-parallelogram polyominoes and their relation to tree-like structures.
• To investigate the existence and properties of minimal and canonical $m$-bases for polyomino and permutation classes.
• To extend known results on Wilf-equivalences in permutation patterns to the context of polyomino classes via submatrix avoidance.

Proposed method

• Define polyomino classes through avoidance of specific submatrices, using the $m$-basis and $p$-basis formalism to characterize them.
• Apply the Schützenberger methodology (DSV) to derive functional equations for generating functions of $k$-parallelogram polyominoes.
• Construct a bijection between $k$-parallelogram polyominoes and planted plane trees, linking the $k$-convexity degree to tree height.
• Use generating functions and recurrence relations to enumerate $k$-parallelogram polyominoes, deriving a closed-form formula.
• Translate permutation pattern avoidance into submatrix avoidance in binary matrices to define and analyze polyomino classes.
• Employ the Stanley-Wilf-Marcus-Tardos theorem as a theoretical benchmark to assess density properties of polyomino classes.

Experimental results

Research questions

• RQ1 How can polyomino classes be systematically defined and characterized using submatrix avoidance in binary matrices?
• RQ2 What is the relationship between the $p$-basis of a permutation class and its corresponding $m$-basis in the context of polyomino classes?
• RQ3 Can the generating function for $k$-parallelogram polyominoes be derived explicitly, and what is its combinatorial interpretation?
• RQ4 Under what conditions is the $p$-basis of a class also a minimal $m$-basis, and how can such bases be computed?
• RQ5 Do polyomino classes exhibit density properties analogous to the Stanley-Wilf theorem for permutation classes?

Key findings

• The class of $k$-parallelogram polyominoes is enumerated via a generating function derived from a functional equation, with a closed-form formula provided.
• A bijective correspondence is established between $k$-parallelogram polyominoes and planted plane trees, where the $k$-convexity degree corresponds to the height of the tree.
• The family of snake-shaped polyominoes is shown to be a polyomino class defined by avoidance of specific submatrices.
• Hollow stack polyominoes and rectangles with rectangular holes are characterized as polyomino classes through explicit $p$-bases.
• The $p$-basis of a polyomino class can be derived from any $m$-basis, and the canonical $m$-basis provides a minimal representation for such classes.
• The submatrix avoidance approach enables the derivation of infinitely many Wilf-equivalences in permutation classes, extending known results in the literature.