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Ascending Convex Polyominoes

Nicholas Beaton, Simone Rinaldi

Abstract

Convex polyominoes can be refined according to the number of direction changes in monotone paths connecting pairs of cells, leading to the notion of $k$-convexity. In particular, the cases $k=1$ and $k=2$ correspond to $L$-convex and $Z$-convex polyominoes, two well-studied subclasses of convex polyominoes, with intermediate families such as centered and $4$-stack polyominoes. These families exhibit remarkably different combinatorial behaviours, suggesting that geometric constraints have a strong impact on the nature of the generating function: $L$-convex and centered polyominoes possess rational generating functions and growth of order $(2+\sqrt{2})^n$ and $4^n$, respectively, while $Z$-convex, 4-stack, and convex polyominoes have algebraic functions and asymptotics of order $n4^n$, $\sqrt{n}\,4^n$, and $n4^n$ respectively. In this paper we investigate the structure of $Z$-convex polyominoes by introducing a refinement based on the NW- and NE-convexity degrees, which yields a decomposition into three disjoint subclasses $C(1,2)$, $C(2,1)$, and $C(2,2)$. To enumerate these families we introduce ascending polyominoes, admitting a simple geometric characterization, and construct a generating tree that leads to functional equations for the corresponding generating functions. By solving these equations we obtain explicit algebraic generating functions and the asymptotic growth for all the subclasses.

Ascending Convex Polyominoes

Abstract

Convex polyominoes can be refined according to the number of direction changes in monotone paths connecting pairs of cells, leading to the notion of -convexity. In particular, the cases and correspond to -convex and -convex polyominoes, two well-studied subclasses of convex polyominoes, with intermediate families such as centered and -stack polyominoes. These families exhibit remarkably different combinatorial behaviours, suggesting that geometric constraints have a strong impact on the nature of the generating function: -convex and centered polyominoes possess rational generating functions and growth of order and , respectively, while -convex, 4-stack, and convex polyominoes have algebraic functions and asymptotics of order , , and respectively. In this paper we investigate the structure of -convex polyominoes by introducing a refinement based on the NW- and NE-convexity degrees, which yields a decomposition into three disjoint subclasses , , and . To enumerate these families we introduce ascending polyominoes, admitting a simple geometric characterization, and construct a generating tree that leads to functional equations for the corresponding generating functions. By solving these equations we obtain explicit algebraic generating functions and the asymptotic growth for all the subclasses.

Paper Structure

This paper contains 19 sections, 8 theorems, 38 equations, 7 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Basic definitions and properties
  3. Convexity degree and $k$-convex polyominoes
  4. Ascending convex polyominoes
  5. Generation of ascending polyominoes
  6. Labeling ascending polyominoes
  7. Operations on centered ascending polyominoes
  8. The Nc operation.
  9. Operations on non-centered ascending polyominoes
  10. A generating tree for ascending polyominoes
  11. The productions of ascending polyominoes
  12. Solving the functional equations
  13. Rectangular Case
  14. Non-rectangular case
  15. Generating functions of some families of ascending polyominoes
  16. ...and 4 more sections

Key Result

Proposition 2.1

Let $P$ be a convex polyomino such that $D_\mathrm{NW}(P) < D_\mathrm{NE}(P)$. If $D_\mathrm{NE}(P) > 2$, then $D_\mathrm{NW}(P) =1$.

Figures (7)

  • Figure 1: (a) stack polyomino; (b) $L$-convex polyomino; (c) centered polyomino; (d) $Z$-convex polyomino; (e) 4-stack polyomino.
  • Figure 2: The two possible configurations for the points $\alpha,\beta,\gamma,\delta$.
  • Figure 3: Graphical representation of the proof of Proposition \ref{['prop:inclusion']}.
  • Figure 4: (a),(b) Convex polyomino in $\mathcal{C}(3,1)$; (c) Centered ascending polyomino; (d) Non-centered polyomino in $\mathcal{C}(2,2)$. (e) Non-centered ascending polyomino in $\mathcal{C}(2,1)$.
  • Figure 5: Centered ascending polyominoes: (a) non-rectangular, $b>1$, $r>0$; (b) rectangular, $b>1$, $r>0$; (b) flipped stack: $w=\ell(P)$, $r=0$; (c) non-rectangular, $b>1$, $w=0$, $r=0$.
  • ...and 2 more figures

Theorems & Definitions (10)

  • Proposition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • Proposition 2.4
  • Theorem 3.2
  • proof
  • Theorem 6.1
  • Theorem 6.2
  • Theorem 6.3