Integer Partitions, Tilings of 2D-gons and Lattices

Tilings of Zonotopes and Lattices

In this paper, we study two kinds of combinatorial objects, generalized integer partitions and tilings of two dimensional zonotopes, using dynamical systems and order theory. We show that the sets of partitions ordered with a simple dynamics, have the distributive lattice structure. Likewise, we show that the set of tilings of zonotopes, ordered with a simple and classical dynamics, is the disj...

Partitions Excluding Specific Polygonal Numbers As Parts

A partition λ of the nonnegative integer n is a sequence of nonnegative integers λ1 ≥ λ2 ≥ · · · ≥ λr with λ1 + λ2 + · · · + λr = n. Each value λi, 1 ≤ i ≤ r, is called a part of the partition. In this note, we consider partitions of n with parts related to k-gonal numbers for some fixed integer k ≥ 3. Various works have appeared involving partitions into polygonal parts. For example, M. D. Hir...

Sl2(z)-tilings of the Torus, Coxeter-conway Friezes and Farey Triangulations

The notion of SL2-tiling is a generalization of that of classical Coxeter-Conway frieze pattern. We classify doubly antiperiodic SL2-tilings that contain a rectangular domain of positive integers. Every such SL2-tiling corresponds to a pair of frieze patterns and a unimodular 2 × 2-matrix with positive integer coefficients. We relate this notion to triangulated n-gons in the Farey graph.

Pentaheptite Modifications of the Graphite Sheet

Pentaheptites (three-coordinate tilings of the plane by pentagons and heptagons only) are classified under the chemically motivated restriction that all pentagons occur in isolated pairs and all heptagons have three heptagonal neighbors. They span a continuum between the two lattices exemplified by the boron nets in ThMoB4 (cmm) and YCrB4 (pgg), in analogy with the crossover from cubic-close-pa...

Generalized Integer Partitions , Tilings of