Compression Theorems for Periodic Tilings and Consequences

In Proofs that Really Count [2], Benjamin and Quinn have used "square and domino tiling" interpretation to provide tiling proofs of many Fibonacci and Lucas formulas. We explore this approach in order to provide tiling proofs of some generalized Fibonacci and Lucas identities.

Cornell University - arXiv, 2018

In two papers, Little and Sellers introduced an exciting new combinatorial method for proving partition identities which is not directly bijective. Instead, they consider various sets of weighted tilings of a 1 × ∞ board with squares and dominoes, and for each type of tiling they construct a generating function in two different ways, which generates a q-series identity. Using this method, they recover quite a few classical q-series identities, but Euler's Pentagonal Number Theorem is not among them. In this paper, we introduce a key parameter when constructing the generating functions of various sets of tilings which allows us to recover Euler's Pentagonal Number Theorem along with an infinite family of generalizations.

International Journal of Computational Geometry & Applications, 2011

A set of natural numbers tiles the plane if a square-tiling of the plane exists using exactly one square of sidelength n for every n in the set. From Ref. 8 we know that ℕ itself tiles the plane. From that and Ref. 9 we know that the set of even numbers tiles the plane while the set of odd numbers doesn't. In this paper we explore the nature of this property. We show, for example, that neither tiling nor non-tiling is preserved by superset. We show that a set with one or three odd numbers may tile the plane—but a set with two odd numbers can't. We find examples of both tiling and non-tiling sets that can be partitioned into tiling sets, non-tiling sets or a combination. We show that any set growing faster than the Fibonacci numbers cannot tile the plane.

2015

We relate a balancing property of letters for bi-infinite sequences to the invariance of the resulting 1-dimensional tiling dynamics under changes in the lengths of the tiles. If the language of the sequence space is finitely balanced, then all length changes in the corresponding tiling space result in topological conjugacies, up to an overall rescaling.

2022

In this paper we explore two types of tilings of a honeycomb strip and derive some closed form formulas for the number of tilings. Furthermore, we obtain some new identities involving tribonacci numbers, Padovan numbers and Narayana’s cow sequence and provide combinatorial proofs for several known identities about those numbers.

Theoretical Computer Science, 2013

We study the combinatorial properties and the problem of generating exhaustively double square tiles, i.e. polyominoes yielding two distinct periodic tilings by translated copies such that every polyomino in the tiling is surrounded by exactly four copies. We show in particular that every prime double square tile may be obtained from the unit square by applying successively some invertible operators on double squares. As a consequence, we prove a conjecture of Provençal and Vuillon [17] stating that these polyominoes are invariant under rotation by angle π.

Integers

We provide tiling proofs for some relations between Fibonacci and Lucas numbers, as requested by Benjamin and Quinn in their text, Proofs that Really Count. Extending our arguments yields Gibonacci generalizations of these identities.

Journal of Combinatorial Theory, Series A, 1990

Discrete & Computational Geometry, 2011

An n-dimensional cross consists of 2n + 1 unit cubes: the "central" cube and reflections in all its faces. A tiling by crosses is called a Z-tiling if each cross is centered at a point with integer coordinates. Periodic tilings of R n by crosses have been constructed by several authors for all n ∈ N. No non-periodic tiling of R n by crosses has been found so far. We prove that if 2n + 1 is not a prime, then the total number of non-periodic Z-tilings of R n by crosses is 2 ℵ 0 while the total number of periodic Z-tilings is only ℵ 0. In a sharp contrast to this result we show that any two tilings of R n , n = 2, 3, by crosses are congruent. We conjecture that this is the case not only for n = 2, 3, but for all n where 2n + 1 is a prime.

The Ramanujan Journal, 2019

In two papers, Little and Sellers introduced an exciting new combinatorial method for proving partition identities which is not directly bijective. Instead, they consider various sets of weighted tilings of a 1 × ∞ board with squares and dominoes, and for each type of tiling they construct a generating function in two different ways, which generates a q-series identity. Using this method, they recover quite a few classical q-series identities, but Euler's Pentagonal Number Theorem is not among them. In this paper, we introduce a key parameter when constructing the generating functions of various sets of tilings which allows us to recover Euler's Pentagonal Number Theorem along with an infinite family of generalizations.