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Enumeration of paths in a hexagonal circle packing

Jean-Luc Baril, José Luis Ramí rez

TL;DR

This work analyzes paths in the hexagonal circle packing by enumerating them with respect to width, height, number of steps, area, and kissing number. It develops and solves functional equations via the kernel method to produce closed multivariate generating functions, and it establishes rich bijections with skew Dyck paths and constrained/peakless Motzkin paths. The counting arrays for several enumerations are shown to be Riordan arrays, with explicit A- and Z-sequences, and continued-fraction expansions are derived for area and kissing-number enumerators. The results connect to RNA secondary structures, Catalan and Schröder numbers, and offer a framework for further combinatorial and geometric explorations of packing paths and their self-avoiding variants.

Abstract

We investigate paths in the hexagonal circle packing and enumerate them with respect to width, height, number of steps, area, and kissing number. Functional equations and the kernel method yield closed bivariate generating functions together with coefficient formulas and asymptotics. We establish bijections with skew Dyck paths, constrained Motzkin paths, and peakless Motzkin paths, and show that several of the associated counting arrays are Riordan arrays. Continued-fraction expansions for the area and kissing-number enumerators are also obtained.

Enumeration of paths in a hexagonal circle packing

TL;DR

This work analyzes paths in the hexagonal circle packing by enumerating them with respect to width, height, number of steps, area, and kissing number. It develops and solves functional equations via the kernel method to produce closed multivariate generating functions, and it establishes rich bijections with skew Dyck paths and constrained/peakless Motzkin paths. The counting arrays for several enumerations are shown to be Riordan arrays, with explicit A- and Z-sequences, and continued-fraction expansions are derived for area and kissing-number enumerators. The results connect to RNA secondary structures, Catalan and Schröder numbers, and offer a framework for further combinatorial and geometric explorations of packing paths and their self-avoiding variants.

Abstract

We investigate paths in the hexagonal circle packing and enumerate them with respect to width, height, number of steps, area, and kissing number. Functional equations and the kernel method yield closed bivariate generating functions together with coefficient formulas and asymptotics. We establish bijections with skew Dyck paths, constrained Motzkin paths, and peakless Motzkin paths, and show that several of the associated counting arrays are Riordan arrays. Continued-fraction expansions for the area and kissing-number enumerators are also obtained.

Paper Structure

This paper contains 11 sections, 23 theorems, 133 equations, 11 figures.

Table of Contents

  1. Introduction and motivation
  2. Partial packing of a given width
  3. A bijection with the skew Dyck paths
  4. A connection with Riordan arrays
  5. Partial packing paths with a given number of steps
  6. A bijection with certain Motzkin paths.
  7. A connection with Riordan arrays
  8. Packing paths with a given area
  9. Packing paths with a given kissing number
  10. Inchworm paths
  11. Going further

Key Result

Theorem 1

We have Finally, the bivariate generating function $S(x,u)$, where the coefficient of $x^nu^k$ is the number of paths of width $n$ ending at height $k$, satisfies

Figures (11)

  • Figure 1: The hexagonal circle packing and the associated hexagon--triangle tiling.
  • Figure 2: On the left, the partial packing path $U\overline{U}\overline{D}\overline{F}FDUF\overline{F}\overline{U}\overline{D}\overline{F}\overline{U}UF\overline{F}FD\overline{D}\overline{F}\overline{U}$ with 21 steps, ending at height 2 and abscissa 30. On the right, the packing path $U\overline{U}\overline{D}\overline{F}FDUF\overline{F}\overline{U}\overline{D}\overline{F}\overline{U}UF\overline{F}FD\overline{D}D$ with 20 steps, ending on the $x$-axis at abscissa 28, area 9 and kissing number 8.
  • Figure 3: The 10 paths ending on the $x$-axis at abscissa 8.
  • Figure 4: The packing path $P=U\overline{U}~UFD~\overline{D}D\overline{F}U\overline{D}DUF\overline{F}FD$ and its image by $f$: $f(P)=uu\textcolor{red}{d'}uud\ell dud$.
  • Figure 5: The 5 paths with 7 steps (ending on the $x$-axis at abscissa $4$).
  • ...and 6 more figures

Theorems & Definitions (28)

  • Theorem 1
  • Corollary 1
  • Theorem 2
  • proof
  • Corollary 2
  • Corollary 3
  • Corollary 4
  • Proposition 1
  • Theorem 3
  • Corollary 5
  • ...and 18 more