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Generalized Integer Partitions, Tilings of Zonotopes and Lattices

M. Latapy

TL;DR

The paper addresses the interplay between generalized integer partitions and tilings of 2D zonotopes, using dynamical systems and order theory to reveal lattice structures. It proves that partition spaces $P(G,h)$ form distributive lattices with coordinate-wise infimum and supremum, and that hypersolid and linear partitions correspond to well-known ideal lattices, connecting to $P(H(d,s,h))$ and $\mathbb{N}^d$. For tilings, the set of $D\rightarrow 2$ tilings $T(Z,D,2)$ decomposes into a disjoint union of distributive lattices, and deleting a fixed family yields a quotient isomorphic to a lower-dimensional tiling set $T(Z',D-1,2)$, enabling recursive structure and potential algorithms for flip sequences. These results provide a unifying framework linking partitions and zonotopal tilings and suggest directions for higher-dimensional generalizations and efficient computations in tiling dynamics.

Abstract

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 disjoint union of distributive lattices which we describe. We also discuss the special case of linear integer partitions, for which other dynamical systems exist. These results give a better understanding of the behaviour of tilings of zonotopes with flips and dynamical systems involving partitions.

Generalized Integer Partitions, Tilings of Zonotopes and Lattices

TL;DR

The paper addresses the interplay between generalized integer partitions and tilings of 2D zonotopes, using dynamical systems and order theory to reveal lattice structures. It proves that partition spaces form distributive lattices with coordinate-wise infimum and supremum, and that hypersolid and linear partitions correspond to well-known ideal lattices, connecting to and . For tilings, the set of tilings decomposes into a disjoint union of distributive lattices, and deleting a fixed family yields a quotient isomorphic to a lower-dimensional tiling set , enabling recursive structure and potential algorithms for flip sequences. These results provide a unifying framework linking partitions and zonotopal tilings and suggest directions for higher-dimensional generalizations and efficient computations in tiling dynamics.

Abstract

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 disjoint union of distributive lattices which we describe. We also discuss the special case of linear integer partitions, for which other dynamical systems exist. These results give a better understanding of the behaviour of tilings of zonotopes with flips and dynamical systems involving partitions.

Paper Structure

This paper contains 8 sections, 5 theorems, 7 equations, 8 figures.

Table of Contents

  1. Preliminaries
  2. The lattices of integers partitions
  3. Generalized integer partitions
  4. Hypersolid and Linear partitions
  5. The lattice of the tilings of a zonotope
  6. Conclusion and perspectives
  7. Erratum
  8. Acknowledgments

Key Result

Theorem 1

Given a partition problem $(G=(V,E),h)$, the set $P(G,h)$ equipped with the order induced by the transition rule is a distributive lattice. Moreover, the infimum (resp. supremum) of two given partitions $a$ and $b$ in this set is the partition $c$ (resp. $d$) defined by:

Figures (8)

  • Figure 1: Example of the dynamics with the partition problem $(G_1,2)$: the first partitions of the set $P(G_1,2)$ with the possible transitions. We show in each vertex the number of grains it contains. Notice that we do not display all the reachable partitions in this diagram, but only the first ones.
  • Figure 2: The solutions of $(G,h)$ are nothing but the ideals of $G \times \lbrace 1,2,\dots,h \rbrace$. Here, we show a graph $G$ (left) with vertices $\lbrace a,b,c,d,e\rbrace$ and the product $G \times \lbrace 1,2,3,4\rbrace$ (right), which is equivalent to the partition problem $(G,4)$. We show the ideal equivalent to the partition displayed on the graph.
  • Figure 3: The two evolution rules of the dynamical system defined by Brylawski.
  • Figure 4: On the left, the diagram of the lattice $L_B(7)$, and on the right the diagram of $SPM(7)$. We showed the representation by piles of grains, and we displayed on each edge the column from which the grain falls during the corresponding transition.
  • Figure 5: Examples of tilings. Left : a $3\rightarrow 2$ tiling. Right : a $4\rightarrow 2$ tiling. Notice that if we delete the shaded tiles in the $4\rightarrow 2$ tiling then we obtain a $3\rightarrow 2$ tiling.
  • ...and 3 more figures

Theorems & Definitions (5)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • Lemma 2
  • Lemma 3