^{1}

^{1}

Let
*T*
_{n }be the set of ribbon L-shaped n-ominoes for some n
≥4 even, and let
*T ^{+}*

_{n}be

*T*

_{n}with an extra 2 x 2 square. We investigate signed tilings of rectangles by

*T*

_{n}and

*T*

^{+}_{n}. We show that a rectangle has a signed tiling by

*T*

_{n}if and only if both sides of the rectangle are even and one of them is divisible by n, or if one of the sides is odd and the other side is divisible by . We also show that a rectangle has a signed tiling by

*T*

^{+}_{n, }n ≥6 even, if and only if both sides of the rectangle are even, or if one of the sides is odd and the other side is divisible by . Our proofs are based on the exhibition of explicit Gr Öbner bases for the ideals generated by polynomials associated to the tiling sets. In particular, we show that some of the regular tiling results in Nitica, V. (2015) Every tiling of the first quadrant by ribbon L n-ominoes follows the rectangular pattern. Open Journal of Discrete Mathematics, 5, 11-25, cannot be obtained from coloring invariants.

In this article, we study tiling problems for regions in a square lattice by certain symmetries of an L-shaped polyomino. Polyominoes were introduced by Golomb in [

A ribbon polyomino [

Related papers are [

The results in [

Signed tilings (see [

stronger then coloring arguments. By looking at signed tilings of rectangles by

A useful tool in the study of signed tilings is a Gröbner basis associated to the polynomial ideal generated by the tiling set. See Bodini and Nouvel [

Signed tilings by ribbon L n-ominoes, n odd are studied in [

The main results of the paper are the following:

Theorem 1. A rectangle can be signed tiled by

Theorem 1 is proved in Section 5, after finding a Gröbner basis for

Theorem 2. A rectangle can be signed tiled by

Theorem 2 is proved in Section 6, after finding a Gröbner basis for

Due to the Gröbner basis that we exhibit for

Proposition 3. A k-inflated copy of the ribbon L n-omino,

The proof of Proposition 3 is shown in Section 7.

Barnes [

Theorem 4. If complex number weights are used, a rectangle can be signed tiled by

Theorem 4 is proved in Section 8. A Gröbner basis for the tiling set helps even if Barnes method is used.

Theorem 5. If complex number weights are used, a rectangle can be signed tiled by

Theorem 5 is proved in Section 9. It is not clear to us if last statement in Theorem 4 implies Theorem 1 and if Theorem 5 implies Theorem 2. Guided by the work here, we conclude that Gröbner basis method for solving signed tiling problems with integer weights is sometimes more versatile and leads to stronger results then Barnes method.

The methods we use in this paper are well known when applied to a particular tiling problem. Here we apply them uniformly to solve an infinite collection of problems. Our hope was to see some regularity in the Gröbner bases associated to other infinite families of tiling sets, such as the family

Let

For

Definition 1. Let

If

Definition 2. A D-Gröbner basis is a finite set G of

Proposition 6. Let G be a finite set of

1) G is a Gröbner basis.

2) Every

We observe, nevertheless, that if R is only a (PID), the normal form associated to a polynomial f by a finite set G of

We introduce now the notions of S-polynomial and G-polynomial that allows to check if a given finite set G of

Definition 3. Let

If

Theorem 7. Let G be a finite set of

Assume now that R is an Euclidean domain with unique reminders (see page 463 [

Definition 4. Let

Proposition 8. E-reduction extends D-reduction, i.e., every D-reduction step in an E-reduction step.

Theorem 9. Let R be an Euclidean domain with unique reminders, and assume G subset of

1)

2) E-reduction modulo G has unique normal forms.

The following result connects signed tilings and Gröbner bases. See [

Theorem 10. A polyomino P admits a signed tiling by translates of prototiles

We show first Gröbner bases for the ideals generated by

Proposition 11. The polynomials

Proof. The polynomials corresponding to the tiles in

It remains to show that the S-polynomial associated to

Proposition 12. A Gröbner basis for the ideal of polynomials generated by

Proof. The polynomials associated to

Similar to what is done in [

For the rest of this section

We show that a Gröbner basis for the ideal generated by the polynomials in Formula (8) is given by:

where

Proposition 13. The polynomials

Proof. Due to the symmetry, we only show that

We show how to build

The steps of a geometric construction for k odd are shown in

will translate the tile

The steps of a geometric constructions for k even are shown in

We show how to build

The steps of a geometric constructions for k odd are shown in

several times multiples of

Proposition 14. The polynomials

Proof. Due to the symmetry, it is enough to show that

Proposition 15. The sets

Proof. This follows from Propositions 13, 14.

Proposition 16. One has the following formulas:

where

Proof. We observe that we can always choose one of the coefficients

The D-reduction of

The D-reduction of

From Step 3 to Step 4 we subtract

The D-reduction of

From Step 1 to Step 2 we subtract

The D-reduction of

To reach Step 1, we subtract

odd and

The D-reduction of

The D-reduction of

We consider first the case

Proposition 17. A Gröbner basis for the ideal generated by

Proof. The pictures for tiles corresponding to the basis are shown in

thus the Gröbner basis can be generated by the polynomials in

thus

The S-polynomial

Let now

As

Lemma 18. The polynomial

Proof. First produce

Expanding Formula (31) gives a telescopic sum that reduces to

Proposition 19. The polynomials

Proof. We showed in Formula (30) how to generate

By the symmetry of

Finally,

Lemma 20. The polynomials

Proof. We show below independently in Proposition 21 that

Proposition 21. The members of

Proof. One has after calculations:

To obtain

By Lemma 18, this tile may be transformed into

The polynomial

By symmetry, the same process will change

Then, multiply by

Once again, symmetry gives us a procedure for

Proposition 22. The sets

Proof. This follows from Propositions 19, 20.

Proposition 23. We have the following formulas:

which are given by D-reductions. Therefore,

Proof. We start with

The reader may easily check that the given reductions are valid for these S-polynomials.

The case

In what follows the signed tile

Multiplying the polynomial associated to the rectangle by

Reducing further the configurations in

1) The polynomial

2) The extra tiles B that appear while doing tile arithmetic for 1), including those from

If

We discuss first 1) and show that it is true when p or q is divisible by n. Then, assuming this condition satisfied, we discuss 2).

1) Let

If r is odd, one has the sequence of remainders, each remainder written in a separate pair of parentheses:

If

If r is even, one has the sequence of remainders, each remainder written in a separate pair of parentheses:

If

2) We assume now that n divides p or q and count the extra tiles B that appears. They are counted by the coefficients of the quotient, call it

We use the equation relating the derivatives:

Note that

Differentiating the equation of

While computing

We need the following formulas:

Case A. One has:

The number of extra B tiles is

to be a multiple of

Case B. One has:

The number of extra B tiles is

Case C. One has:

The number of extra B tiles is

Let

Assume now

If k is even, finding a signed tiling for a -inflated copy of the Ln-omino can be reduced, via reductions by

In this section we give a proof of Theorem 4 following a method of Barnes. We assume familiarity with [

Separating x from

Denote the polynomial on the left hand side by

We show now that I is a radical ideal. We use an algorithm of Seidenberg which can be applied to find the radical ideal of a zero dimensional algebraic variety over an algebraically closed field. See Lemma 92 in [

The ideal generated by

We apply Lemma 3.8 in [

luates to zero on V. If R is a

evaluates to zero on V if and only if one of

For the second statement in Theorem 4 we use the method described in the proof of Theorem 4.2 in [

Let

Assume now

Understanding tilings of rectangles by particular, even simple, polyominoes is a difficult combinatorial problem with a long history. Among the pioneering contributions, we mention those of Klarner [

Similar problems can be studied for parallelograms. As already mentioned in the introduction, the problem of tiling a general parallelogram positioned on a skewed lattice by all symmetries of a single skewed tile that has all sides parallel to the sides of the parallelogram is equivalent to the problem of tiling a rectangle by a polyomino (the straightened tile) allowing only a reduced set of orientations for that polyomino. This new problem seems to be more amenable to a solution and considerable progress has been done in the case of L-shaped n-ominoes of order two in several recent papers of one of the authors and collaborators [

The main contribution of the present paper is a strengthening of the results in [

V. Nitica was partially supported by Simons Foundation Grant 208729. While working on this project, K. Gill was undergraduate student at West Chester University.

Kenneth Gill,Viorel Nitica, (2016) Signed Tilings by Ribbon L n-Ominoes, n Even, via Gröbner Bases. Open Journal of Discrete Mathematics,06,185-206. doi: 10.4236/ojdm.2016.63017