ABSTRACT (original article): Conway and Lagarias observed that a triangular region
T
2
(n)
in a hexagonal lattice admits a signed tiling by 3-in-line polyominoes (tribones) if and only if
n∈{
2
3
d−1,
2
3
d}
d∈ℕ
. We apply the theory of Gröbner bases over integers to show that
T
3
(n)
, a three dimensional lattice tetrahedron of edge-length n, admits a signed tiling by tribones if and only if
n∈{
3
3
d−2,
3
3
d-1,
3
3
d}
d∈ℕ
. More generally we study Gröbner lattice-point enumerators of lattice polytopes and show that they are (modular) quasipolynomials in the case of k-in-line polyominoes. As an example of the “unusual cancelation phenomenon”, arising only in signed tilings, we exhibit a configuration of 15 tribones in the 3-space such that exactly one lattice point is covered by an odd number of tiles. CITATION (original article): M. Muzika Dizdarević, M. Timotijević, R. T. Živaljević, Gröbner Lattice-Point Enumerators and Signed Tiling by k-in-line Polyominoes, Kragujevac Journal of Mathematics, 49(3) (2025), 443–464. https://doi.org/10.46793/KgJMat2503.443D
In a script below are programed the following objects (it should be executed before continuing):
T[n] - pyramidal polynomial of a side-length n;
TriDBoni - tile polynomials of a length 3;
TriDGrebner - Gröbner basis of an ideal generated by TriDBoni;
OstatakDeljenja[n,d] - a remainder of T[n] divided by TriDGrebner;
Prikazi[p] - shows coefficients of a polynomial p(x,y,z) in a matrix form.
In[]:=
T[n_]:=Together
1
(1-x)*(1-y)*(1-z)
+
x^(n+2)
(x-1)*(x-y)*(x-z)
+
y^(n+2)
(y-1)*(y-x)*(y-z)
+
z^(n+2)
(z-1)*(z-x)*(z-y)
​​TriDBoni={1+x+
2
x
,1+y+
2
y
,1+z+
2
z
,
2
x
+xy+
2
y
,
2
x
+xz+
2
z
,
2
y
+yz+
2
z
}​​TriDGrebner=GroebnerBasis[TriDBoni,{x,y,z},CoefficientDomainIntegers]​​OstatakDeljenja[n_]:=Last[PolynomialReduce[T[n],TriDGrebner,{x,y,z},CoefficientDomainIntegers]]​​Prikazi[p_]:=MatrixForm[CoefficientList[p,{y,x,z}]]
Out[]=
{1+x+
2
x
,1+y+
2
y
,1+z+
2
z
,
2
x
+xy+
2
y
,
2
x
+xz+
2
z
,
2
y
+yz+
2
z
}
Out[]=
{9,-3+3z,1+z+
2
z
,-3+3y,-2-y-z+yz,1+y+
2
y
,-3+3x,-2-x-z+xz,-2-x-y+xy,1+x+
2
x
}
Following script shows the output of the function Prikazi[p].
In[]:=
Prikazi[(x^2+1)z+xy^2]
Out[]//MatrixForm=

0
1


0
0


0
1


0
0


0
0


0
0


0
0


1
0


0
0

Below are illustrated polynomials of the set TriDBoni.
In[]:=
Table[TableForm[{TriDBoni[[i]],Prikazi[TriDBoni[[i]]]}],{i,1,6}]
Out[]=

1+x+
2
x
(
(
1
)
(
1
)
(
1
)
)
,
1+y+
2
y
(
1
)
(
1
)
(
1
)
,
1+z+
2
z
1
1
1
,
2
x
+xy+
2
y
(
0
)
(
0
)
(
1
)
(
0
)
(
1
)
(
0
)
(
1
)
(
0
)
(
0
)
,
2
x
+xz+
2
z
0
0
1
0
1
0
1
0
0
,
2
y
+yz+
2
z
0
0
1
0
1
0
1
0
0

Below are illustrated polynomials of the set TriDGrebner.
In[]:=
Table[TableForm[{TriDGrebner[[i]],Prikazi[TriDGrebner[[i]]]}],{i,1,10}]​​
Out[]=

9
(
(
9
)
)
,
-3+3z


-3
3


,
1+z+
2
z
1
1
1
,
-3+3y
(
-3
)
(
3
)
,
-2-y-z+yz

-2
-1


-1
1

,
1+y+
2
y
(
1
)
(
1
)
(
1
)
,
-3+3x
(
(
-3
)
(
3
)
)
,
-2-x-z+xz


-2
-1


-1
1


,
-2-x-y+xy
(
-2
)
(
-1
)
(
-1
)
(
1
)
,
1+x+
2
x
(
(
1
)
(
1
)
(
1
)
)

Next script shows the coefficients of polynomial T[n] for various values of n.
In[]:=
Manipulate[TableForm[{Expand[T[n]],Prikazi[T[n]]}],Column[{​​Control[{{n,3},{1,2,3,4,5,6,7,8,9,10,11,12,13}}],Spacer[20]}],SaveDefinitions->True]
Out[]=
​
n
3
1+x+
2
x
+y+xy+
2
y
+z+xz+yz+
2
z
1
1
1
1
1
0
1
0
0
1
1
0
1
0
0
0
0
0
1
0
0
0
0
0
0
0
0
Next script shows remainders of division of T[n] by TriDGrebner for various values of n.
MatrixForm[Table[{n,Prikazi[OstatakDeljenja[n]]},{n,1,54}]]
1
(
(
1
)
)
2

1
1


1
0


1
0


0
0

3

4
-1


-1
0


-1
0


0
0

4
(
(
2
)
)
5

2
-1


-1
0


-1
0


0
0

6

-1
1


1
0


1
0


0
0

7
(
(
3
)
)
8
(
(
3
)
)
9
(
(
3
)
)
10
(
(
4
)
)
11

4
1


1
0


1
0


0
0

12

-2
-1


-1
0


-1
0


0
0

13
(
(
-4
)
)
14

-4
-1


-1
0


-1
0


0
0

15

2
1


1
0


1
0


0
0

16
(
(
-3
)
)
17
(
(
-3
)
)
18
(
(
-3
)
)
19
(
(
-2
)
)
20

-2
1


1
0


1
0


0
0

21

1
-1


-1
0


-1
0


0
0

22
(
(
-1
)
)
23

-1
-1


-1
0


-1
0


0
0

24

-4
1


1
0


1
0


0
0

25
{}
26
{}
27
{}
28
(
(
1
)
)
29

1
1


1
0


1
0


0
0

30

4
-1


-1
0


-1
0


0
0

31
(
(
2
)
)
32

2
-1


-1
0


-1
0


0
0

33

-1
1


1
0


1
0


0
0

34
(
(
3
)
)
35
(
(
3
)
)
36
(
(
3
)
)
37
(
(
4
)
)
38

4
1


1
0


1
0


0
0

39

-2
-1


-1
0


-1
0


0
0

40
(
(
-4
)
)
41

-4
-1


-1
0


-1
0


0
0

42

2
1


1
0


1
0


0
0

43
(
(
-3
)
)
44
(
(
-3
)
)
45
(
(
-3
)
)
46
(
(
-2
)
)
47

-2
1


1
0


1
0


0
0

48

1
-1


-1
0


-1
0


0
0

49
(
(
-1
)
)
50

-1
-1


-1
0


-1
0


0
0

51

-4
1


1
0


1
0


0
0

52
{}
53
{}
54
{}
By changing the value n, script below shows the reminder of division of T[n] by TriDGrebner.

CITE THIS NOTEBOOK

Gröbner lattice-point enumerators and signed tiling by k-in-line polyominoes​
by M. Muzika Dizdarević, M. Timotijević & R. T. Živaljević
Wolfram Community, STAFF PICKS, September 20, 2024
​https://community.wolfram.com/groups/-/m/t/3278546