Also available at http://amc.imfm.si
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ARS MATHEMATICA CONTEMPORANEA 5 (2012) 107–126
Polyominoes with nearly convex columns:
A semi-directed model
Svjetlan Feretić
Faculty of Civil Engineering, University of Rijeka
Radmile Matejčić 3, 51000 Rijeka, Croatia
Received 30 June 2010, accepted 1 May 2011, published online 12 January 2012
Abstract
Column-convex polyominoes are by now a well-explored model. So far, however, no
attention has been given to polyominoes whose columns can have either one or two con-
nected components. This little known kind of polyominoes seems not to be manageable as
a whole. To obtain solvable models, one needs to introduce some restrictions. This paper
is focused on polyominoes with hexagonal cells. The restrictions just mentioned are semi-
directedness and an upper bound (say m) on the size of the gap within a column. As the
upper bound m grows, the solution of the model tends to break into more and more cases.
We computed the area generating functions for m = 1, m = 2 and m = 3. In this paper,
the m = 1 and m = 2 models are solved in full detail. To keep the size of the paper within
reasonable limits, the result for the m = 3 model is stated without proof. The m = 1,
m = 2 and m = 3 models have rational area generating functions, as column-convex
polyominoes do. (It is practically sure, although we leave it unproved, that the area gen-
erating functions are also rational for m = 4, m = 5,. . . .) However, the growth constants
of the new models are 4.114908 and more, whereas the growth constant of column-convex
polyominoes is 3.863131.
Keywords: Polyomino, hexagonal-celled, nearly convex column, semi-directed, area generating func-
tion.
Math. Subj. Class.: 05B50, 05A15
1 Introduction
The enumeration of polyominoes is a topic of great interest to chemists, physicists and
combinatorialists alike. In chemical terms, any polyomino (with hexagonal cells) is a pos-
sible benzenoid hydrocarbon. In physics, determining the number of n-celled polyominoes
E-mail address: svjetlan.feretic@gradri.hr (Svjetlan Feretić)
Copyright c© 2012 DMFA Slovenije
108 Ars Math. Contemp. 5 (2012) 107–126
is related to the study of two-dimensional percolation phenomena. In combinatorics, poly-
ominoes are of interest in their own right because several polyomino models have good-
looking exact solutions.
Let an be the number of n-celled polyominoes. An exact formula for an seems unlikely
to ever be found. However, there exist notable results on the quantity c = limn→∞ n
√
an.
(This quantity is called the growth constant.) In the case of polyominoes with hexagonal
cells, Vöge and Guttmann [21] gave a rigorous proof that the growth constant c exists and
satisfies the inequality 4.8049 ≤ c ≤ 5.9047 . In the same paper, it is estimated that
c = 5.1831478(17).
In this paper, we are not going to improve the above-stated bounds on c. Instead, we are
going to revisit column-convex polyominoes. The first to study this now-familiar model was
Temperley [20] in 1956. Whether with square cells or with hexagonal cells, column-convex
polyominoes have a rational area generating function. When cells are hexagons, the growth
constant of column-convex polyominoes is 3.863131 (Klarner [17]). This growth constant
remained a record (among polyomino models having reasonably simple exact solutions)
until 1982, when Dhar, Phani and Barma [5] invented directed site animals. Directed site
animals with step-set A = {(1, 0), (0, 1)} can be viewed as a family of polyominoes with
square cells, while directed site animals with step-set B = {(1, 0), (0, 1), (1, 1)} can be
viewed as a family of polyominoes with hexagonal cells. If the step-set is B, the growth
constant of directed site animals is exactly 4. Incidentally, with either of the step-sets A and
B, the area generating function of directed site animals is algebraic (which is rather sur-
prising) and satisfies a simple quadratic equation. Later on, in 2002, Bousquet-Mélou and
Rechnitzer [4] introduced stacked directed animals and multi-directed animals. (By now,
each of those two models has been defined in two equivalent ways. There exist the orig-
inal definitions [4], which use heaps of pieces, but there also exist recent definitions [10],
which do not use heaps of pieces.) Stacked directed animals and multi-directed animals
substantially generalize directed site animals. Whether the cells are squares or hexagons,
the area generating function of stacked directed animals is degree-two algebraic, and the
area generating function of multi-directed animals is not D-finite. When cells are hexagons,
the growth constant of stacked directed animals is exactly 4.5, and the growth constant of
multi-directed animals is 4.587894.
Although descended from directed animals, multi-directed animals are also a superset
of column-convex polyominoes. (To be precise, this holds when cells are hexagons. It is
not quite clear whether multi-directed animals with square cells are a superset of column-
convex polyominoes with square cells.) Before the research presented here, multi-directed
animals were unique in this respect. There existed pretty many polyomino models other
than multi-directed animals, but, to my knowledge, not any of those other models is a su-
perset of column-convex polyominoes. Indeed, stacked directed animals are not a superset
of column-convex polyominoes. James, Jensen and Guttmann’s m-convex polygons [16]
are a superset of convex polyominoes, but are not a superset of column-convex polyomi-
noes. Préa’s prudent polygons [18, 6] are not even a superset of convex polyominoes. (The
growth constant of prudent polygons is just 2 [1], but prudent polygons are nevertheless
very popular in these times [3, 15, 19].)
The aim of this paper is to define a model which (a) generalizes column-convex poly-
ominoes, (b) possesses a reasonably simple area generating function, and (c) possesses a
high growth constant. In view of the facts stated above, we shall have to compete with just
one pre-existing model, namely with multi-directed animals.
S. Feretić: Polyominoes with nearly convex columns: A semi-directed model 109
Figure 1: A hexagonal-celled polyomino.
In this paper, we actually introduce a sequence of generalizations of hexagonal-celled
column-convex polyominoes. Namely, we define level m cheesy polyominoes (m = 1, 2,
3, . . .). At every level, our new model has a rational area generating function. (This is prac-
tically sure, although we do not give a proof.) However, as level increases, those rational
generating functions quickly gain in size. In computing generating functions, we go up
to level 3. The computations are done by using Bousquet-Mélou’s [2] and Svrtan’s [13]
“turbo” version of the Temperley method [20]. The growth constants of level 1, level 2 and
level 3 cheesy polyominoes turn out to be 4.114908, 4.231836 and 4.288631, respectively.
Thus, the growth constants of cheesy polyominoes are not as high as 4.587894, the
growth constant of multi-directed animals. However, especially at level one, cheesy poly-
ominoes have a simple area generating function, which cannot be said about multi-directed
animals. Another good point of cheesy polyominoes is that their definition is short and
immediate.
2 Definitions and conventions
There are three regular tilings of the Euclidean plane, namely the triangular tiling, the
square tiling, and the hexagonal tiling. We adopt the convention that every square or hexag-
onal tile has two horizontal edges. In a regular tiling, a tile is often referred to as a cell. A
plane figure P is a polyomino if P is a union of finitely many cells and the interior of P is
connected. See Figure 1. Observe that, if a union of hexagonal cells is connected, then it
possesses a connected interior as well.
Let P and Q be two polyominoes. We consider P and Q to be equal if and only if there
exists a translation f such that f(P ) = Q.
From now on, we concentrate on the hexagonal tiling. When we write “a polyomino”,
we actually mean “a hexagonal-celled polyomino”.
Given a polyomino P , it is useful to partition the cells of P according to their horizontal
110 Ars Math. Contemp. 5 (2012) 107–126
Figure 2: A column-convex polyomino.
projection. Each block of that partition is a column of P . Note that a column of a polyomino
is not necessarily a connected set. An example of this is the highlighted column in Figure 1.
On the other hand, it may happen that every column of a polyomino P is a connected set.
In this case, the polyomino P is a column-convex polyomino. See Figure 2.
Let a and b be two adjacent columns of a polyomino P . Let the column b be the right
neighbour of the column a. It is certain that column b has at least one edge in common with
column a; otherwise P could not be a connected set. However, there is no guarantee that
every connected component of b has at least one edge in common with a. For example, in
Figure 1, the upper component of the highlighted column has no edge in common with the
previous column.
Suppose that P is such a polyomino that the first (i.e., leftmost) column of P has no
gap and that, in every pair of adjacent columns of P , every connected component of the
right column has at least one edge in common with the left column. Then we say that P is
a rightward-semi-directed polyomino.
A polyomino P is a level m cheesy polyomino if the following holds:
• P is a rightward-semi-directed polyomino,
• every column of P has at most two connected components,
• if a column of P has two connected components, then the gap between the compo-
nents consists of at most m cells.
See Figure 3.
Level one cheesy polyominoes are a subset of level two cheesy polyominoes, level two
cheesy polyominoes are a subset of level three cheesy polyominoes, and so on. As m tends
to infinity, the set of level m cheesy polyominoes tends (in a certain sense) to the set of all
polyominoes which are rightward-semi-directed and are made up of columns with at most
two connected components.
The name “cheesy polyominoes” is intended to suggest that these polyominoes can
have internal holes. At this point, it might be objected that there exists another model,
called directed animals, in which holes (internal and not) occur more freely than in cheesy
S. Feretić: Polyominoes with nearly convex columns: A semi-directed model 111
Figure 3: A level one cheesy polyomino.
Figure 4: The pivot cell.
polyominoes. A column of a directed animal can have any number of holes, and the sizes
of those holes are not subject to any limitations. However, the name of our model has 12
years of tradition1. So, our model will retain the name “cheesy polyominoes” despite the
fact that directed animals are arguably “cheesier”.
Let P be a polyomino with two or more columns. Then we define the pivot cell of
P to be the lower right neighbour of the lowest cell of the second last column of P . See
Figure 4. Observe that the pivot cell of a polyomino P is not necessarily contained in P .
(Figure 4 shows the pivot cells of two column-convex polyominoes. However, the pivot
cell is also defined for polyominoes which are not column-convex and have two or more
columns.)
If a polyomino P is made up of n cells, we say that the area of P is n.
1The contents of this paper exist since 1999. In that year, I defined cheesy polyominoes, explored them,
and presented them at Mathematical Colloquium in Osijek [7], as well as at MATH/CHEM/COMP Course &
Conference in Dubrovnik [8].
112 Ars Math. Contemp. 5 (2012) 107–126
Let a be a column of a polyomino P . By the height of a we mean the number of those
cells which make up a plus the number of those (zero or more) cells which make up the
gaps of a. For example, in Figure 1, the highlighted column has height 7, and the next
column to the left has height 4.
Let R be a set of polyominoes. By the area generating function of R we mean the
formal sum ∑
P∈R
qarea of P .
By the area and last column generating function of R we mean the formal sum∑
P∈R
qarea of P · tthe height of the last column of P .
3 Column-convex polyominoes
The area generating function for column-convex polyominoes is known since 1967 (Klarner
[17]). We shall state that well-known result without proof. The reader need not worry about
this unexplained gap in our presentation. Namely, once the reader gets through the proof
for level one cheesy polyominoes (given in the next section), it will be clear how the afore-
mentioned gap can be bridged.
Let A1 = A1(q) be the area generating function for column-convex polyominoes.
Proposition 3.1. The area generating function for column-convex polyominoes is given by
A1 =
q · (1− q)3
1− 6q + 10q2 − 7q3 + q4
.
Corollary 3.2. The number of n-celled column-convex polyominoes [qn]A1 has the asymp-
totic behaviour
[qn]A1 ∼ 0.188420 · 3.863131n.
Thus, the growth constant of column-convex polyominoes is 3.863131.
4 Level one cheesy polyominoes
Counting level one cheesy polyominoes by area is similar to counting column-convex poly-
ominoes by area.
Let C = C(q, t) be the area and last column generating function for level one cheesy
polyominoes. Let C1 = C(q, 1) and D1 = ∂C∂t (q, 1).
Let T be the set of all level one cheesy polyominoes. We write Tα for the set of level
one cheesy polyominoes which have only one column. Let
Tβ = {P ∈ T \ Tα : the last column of P has no hole,
and the pivot cell of P is contained in P},
Tγ = {P ∈ T \ Tα : the last column of P has no hole,
and the pivot cell of P is not contained in P}, and
Tδ = {P ∈ T \ Tα : the last column of P has a hole}.
S. Feretić: Polyominoes with nearly convex columns: A semi-directed model 113
Figure 5: The last two columns of two elements of Tγ .
The sets Tα, Tβ , Tγ and Tδ form a partition of T . We write Cα, Cβ , Cγ and Cδ to
denote the parts of the series C that come from the sets Tα, Tβ , Tγ and Tδ , respectively.
We have
Cα = qt+ (qt)
2 + (qt)3 + . . . =
qt
1− qt
(4.1)
and
Cβ =
qt
(1− qt)2
· C1. (4.2)
Consider the following situation. A cheesy polyomino P ends with a column I . We are
creating a new column to the right of I , and the result should be an element of Tγ . Then,
whether or not the column I has a hole, we can put the lowest cell of the new column in
exactly m places, where m is the height of I . See Figure 5. Hence
Cγ =
qt
1− qt
·D1. (4.3)
Let us move on to another situation. A cheesy polyomino P ends with a column J .
We are creating a new column to the right of J , and the result should be an element of
Tδ . Then, whether or not the column J has a hole, we can put the hole of the new column
in exactly n − 1 places, where n is the height of J . See Figure 6. The new column is
114 Ars Math. Contemp. 5 (2012) 107–126
Figure 6: The last two columns of two elements of Tδ .
made up of i ∈ {1, 2, 3, . . . } cells lying below the hole, of a hole of height one, and of
j ∈ {1, 2, 3, . . . } cells lying above the hole. Altogether,
Cδ =
qt
1− qt
· t · qt
1− qt
· (D1 − C1) =
q2t3
(1− qt)2
· (D1 − C1). (4.4)
Since C = Cα + Cβ + Cγ + Cδ , Eqs. (4.1), (4.2), (4.3) and (4.4) imply that
C =
qt
1− qt
+
qt
(1− qt)2
· C1 +
qt
1− qt
·D1 +
q2t3
(1− qt)2
· (D1 − C1). (4.5)
Setting t = 1, from Eq. (4.5) we obtain
C1 =
q
1− q
+
q
(1− q)2
· C1 +
q
1− q
·D1 +
q2
(1− q)2
· (D1 − C1).
Differentiating Eq. (4.5) with respect to t and then setting t = 1, we obtain
D1 =
q
1− q
+
q2
(1− q)2
+
[
q
(1− q)2
+
2q2
(1− q)3
]
· C1
+
[
q
1− q
+
q2
(1− q)2
]
·D1 +
[
3q2
(1− q)2
+
2q3
(1− q)3
]
· (D1 − C1).
S. Feretić: Polyominoes with nearly convex columns: A semi-directed model 115
Thus, we have a system of two linear equations in two unknowns, C1 and D1. By
solving the system, we get this proposition.
Proposition 4.1. The area generating function for level one cheesy polyominoes is given
by
C1 =
q(1− 3q + q2)
1− 6q + 8q2 − q3
.
Of course, the denominator of C1 has three complex roots. Those roots are2 r1 =
0.243019, r2 = 0.572771 and r3 = 7.184210. The root with smallest absolute value is
r1 = 0.243019, and 1r1 is equal to 4.114908. Therefore, the coefficient of q
n in C1 (denoted
[qn]C1) has the asymptotic behaviour [qn]C1 ∼ k · 4.114908n, where k is a constant. To
find the value of k, we decompose C1 into partial fractions. The partial fraction involving r1
turns out to be − 0.035038q−0.243019 . In the Taylor series expansion of −
0.035038
q−0.243019 , the coefficient
of qn is equal to 0.144176 · 4.114908n. Thus, k = 0.144176. We have got the following
result.
Corollary 4.2. The number of n-celled level one cheesy polyominoes [qn]C1 has the
asymptotic behaviour
[qn]C1 ∼ 0.144176 · 4.114908n.
Thus, the growth constant of level one cheesy polyominoes is 4.114908. We observe
a considerable increase with respect to 3.863131, the growth constant of column-convex
polyominoes.
5 Level two cheesy polyominoes
In this enumeration, if the last column of a polyomino has two connected components, we
sometimes need to record not only the overall height of the last column, but also the height
of the last column’s upper component and the height of the last column’s lower component.
Hence, in addition to the “old” variables q and t, we introduce two new variables, u and v.
As before, the exponent of q is the area and the exponent of t is the overall height of the
last column3. The exponent of u is the height of the upper component of the last column,
and the exponent of v is the height of the lower component of the last column.
The two main generating functions in this enumeration are E = E(q, t) and G =
G(q, t, u, v). Those generating functions are used for the following purposes:
• E is a generating function for level two cheesy polyominoes whose last column either
has no hole or has a one-celled hole,
• G is a generating function for level two cheesy polyominoes whose last column has
a two-celled hole.
Let E1 = E(q, 1), F0 = ∂E∂t (q, 0), F1 =
∂E
∂t (q, 1), G1 = G(q, 1, 1, 1), H1 =
∂G
∂t (q, 1, 1, 1), I0 =
∂G
∂u (q, 1, 0, 1), and J0 =
∂G
∂v (q, 1, 1, 0).
2Those roots have infinitely many digits. Since here we do not have an infinite amount of space, we shall
round the roots to six decimal places.
3Recall what do we mean by the height of a column: in Figure 1, the highlighted column has height 7, and the
next column to the left has height 4.
116 Ars Math. Contemp. 5 (2012) 107–126
Let U be the set of those level two cheesy polyominoes whose last column either has
no hole or has a one-celled hole. Let V be the set of those level two cheesy polyominoes
whose last column has a two-celled hole. For P ∈ U ∪ V , we define the body of P to be
all of P , except the rightmost column of P .
We are now going to partition the set U into seven subsets and the set V into two
subsets. Let
Uα = {P ∈ U : P has only one column},
Uβ = {P ∈ U \ Uα : the body of P lies in U, the last column of P has no hole,
and the pivot cell of P is contained in P},
Uγ = {P ∈ U \ Uα : the body of P lies in U, the last column of P has no hole,
and the pivot cell of P is not contained in P},
Uδ = {P ∈ U \ Uα : the body of P lies in U, and the last column of P
has a hole},
U = {P ∈ U \ Uα : the body of P lies in V , the last column of P has no hole,
and the pivot cell of P is contained in P},
Uζ = {P ∈ U \ Uα : the body of P lies in V , the last column of P has no hole,
and the pivot cell of P is not contained in P}, and
Uη = {P ∈ U \ Uα : the body of P lies in V , and the last column of P
has a hole}.
Let
Vα = {P ∈ V : the body of P lies in U}, and
Vβ = {P ∈ V : the body of P lies in V }.
It is clear that the sets Uα, Uβ , . . . , Uη form a partition of U , and that the sets Vα and
Vβ form a partition of V . We shall write Eα, Eβ , . . . , Eη for the parts of the series E that
come from the sets Uα, Uβ , . . . , Uη , respectively. Also, we shall write Gα and Gβ for the
parts of the series G that come from the sets Vα and Vβ , respectively.
As in Section 4, we have
Eα =
qt
1− qt
, (5.1)
Eβ =
qt
(1− qt)2
· E1, (5.2)
Eγ =
qt
1− qt
· F1, (5.3)
Eδ =
q2t3
(1− qt)2
· (F1 − E1), (5.4)
E =
qt
(1− qt)2
·G1. (5.5)
S. Feretić: Polyominoes with nearly convex columns: A semi-directed model 117
Figure 7: (a) For this choice of the lowest cell, the new column must be at least two cells
high. (b) For this choice of the lowest cell, the new column does not have to be at least two
cells high.
The functional equation for Eζ is more interesting. Let P be an element of V , let I
be the last column of P , and let m be the height of I . Suppose that we are creating a new
column to the right of I , and that the result should be an element of Uζ . To the right of I
there is one cell (say c) which shares an edge with each of the two cells forming the hole
of I . If we choose c as the lowest cell of the new column, then the new column will have
to be at least two cells high. Otherwise P ∪ (the new column) will not be a polyomino.
See Figure 7. In addition to c, there are m− 1 other choices for the lowest cell of the new
column. For each of these m−1 choices, P ∪(the new column) is a polyomino regardless
of how many cells the new column has. Altogether, we have
Eζ =
q2t2
1− qt
·G1 +
qt
1− qt
· (H1 −G1). (5.6)
We proceed to Eη . Let P be an element of V , let J be the last column of P , and let
n be the height of J . Suppose that we are creating a new column to the right of J , and
118 Ars Math. Contemp. 5 (2012) 107–126
Figure 8: (a) If the one-celled hole is in this position, the upper component of the new
column must have at least two cells. (b) If the one-celled hole is in this position, the upper
and lower components of the new column can be of any sizes.
that the result should be an element of Uη . One of the choices for the hole of the new
column is the upper right neighbour of the top cell of the lower component of J . For this
choice, the upper component of the new column has to have at least two cells. Otherwise
P ∪ (the new column) is not a polyomino. See Figure 8. We can also choose the hole of
the new column as the lower right neighbour of the bottom cell of the upper component of
J . Then, in order for P ∪ (the new column) to be a polyomino, the lower component of
the new column has to have at least two cells. In addition to the two ways just considered,
there exist n− 3 other ways to choose the hole of the new column. For each of those n− 3
ways, P ∪ (the new column) is a polyomino regardless of the sizes of the upper and lower
components of the new column. Thus, we have
Eη =
2q3t4
(1− qt)2
·G1 +
q2t3
(1− qt)2
· (H1 − 3G1). (5.7)
Of course, the series Eα, Eβ , . . . , Eη sum up to E. Therefore, the summation of Eqs.
S. Feretić: Polyominoes with nearly convex columns: A semi-directed model 119
(5.1)–(5.7) gives
E =
qt
1− qt
+
qt
(1− qt)2
· E1 +
qt
1− qt
· F1 +
q2t3
(1− qt)2
· (F1 − E1)
+
qt
(1− qt)2
·G1 +
q2t2
1− qt
·G1 +
qt
1− qt
· (H1 −G1)
+
2q3t4
(1− qt)2
·G1 +
q2t3
(1− qt)2
· (H1 − 3G1). (5.8)
We also need to establish a functional equation for G. Let P be an element of U and let
c be a column with a two-celled hole. Suppose that we want to glue c to P so that P ∪c lies
in Vα, and so that P and c are the body and the last column of P ∪ c, respectively. In how
many ways P and c can be glued together? In principle, the number of ways is (the height
of the last column of P ) minus two. See Figure 9. However, if P ends with a one-celled
column, then we can glue c to P in zero ways, and not in minus one ways. Thus, we have
Gα =
q2t4uv
(1− qtu)(1− qtv)
· (F1 − 2E1 + F0). (5.9)
In the case of Gβ , it is convenient to use overcounting. That is, we are going to “mis-
takenly” count too much, and then subtract the parts which do not belong. Let P be an
element of V and let c be a column with a two-celled hole. Suppose that we want to glue c
to P so that P ∪ c lies in Vβ , and so that P and c are the body and the last column of P ∪ c,
respectively. In how many ways P and c can be glued together? First, there are (the height
of the last column of P ) minus two ways to satisfy these two necessary conditions:
• the bottom cell of the upper component of c is either identical with or lies lower than
the upper right neighbour of the top cell of the last column of P , and
• the top cell of the lower component of c is either identical with or lies higher than
the lower right neighbour of the bottom cell of the last column of P .
See Figure 10.
So, if there were no special cases, then Gβ would be equal to
q2t4uv
(1− qtu)(1− qtv)
· (H1 − 2G1). (5.10)
However, special cases do exist. There are two of them:
1. The upper component of the last column of P ∈ V has at least two cells and the
lower component of the two-component column c has just one cell.
2. The lower component of the last column of P ∈ V has at least two cells and the
upper component of the two-component column c has just one cell.
In case 1, it is (so to speak) dangerous to glue c to P in such a way that the one-
celled lower component of c becomes a common neighbour of the two cells which form
the hole of the last column of P . This dangerous operation produces an object which is not
a polyomino and hence does not lie in Vβ .
120 Ars Math. Contemp. 5 (2012) 107–126
Figure 9: The last two columns of two elements of Vα.
S. Feretić: Polyominoes with nearly convex columns: A semi-directed model 121
Figure 10: The last two columns of five elements of Vβ .
122 Ars Math. Contemp. 5 (2012) 107–126
In case 2, it is dangerous to glue c to P in such a way that the one-celled upper com-
ponent of c becomes a common neighbour of the two cells which form the hole of the last
column of P . Again, the dangerous operation produces an object which is not a polyomino
and hence does not lie in Vβ .
Now, Eq. (5.10) is actually a generating function for the union of Vβ with the set of
objects produced by the two dangerous operations. The generating function for the objects
produced by the first dangerous operation is
q2t4uv
1− qtu
· (G1 − I0). (5.11)
The generating function for the objects produced by the second dangerous operation is
q2t4uv
1− qtv
· (G1 − J0). (5.12)
Subtracting Eqs. (5.11) and (5.12) from Eq. (5.10), we obtain
Gβ =
q2t4uv
(1− qtu)(1− qtv)
·(H1−2G1)−
q2t4uv
1− qtu
·(G1−I0)−
q2t4uv
1− qtv
·(G1−J0). (5.13)
Since G = Gα +Gβ , Eqs. (5.9) and (5.13) imply that
G =
q2t4uv
(1− qtu)(1− qtv)
· (F1 − 2E1 + F0 +H1 − 2G1)
− q
2t4uv
1− qtu
· (G1 − I0)−
q2t4uv
1− qtv
· (G1 − J0). (5.14)
Using the computer algebra system Maple, from Eqs. (5.8) and (5.14) we obtained a
system of seven linear equations in seven unknowns: E1, F0, F1, G1, H1, I0 and J0.
Rather than write down these seven equations (some of which are a bit cumbersome), here
below we give a list of recipes. Recipe no. k (k = 1, 2, . . . , 7) tells how to obtain the kth
equation of the linear system.
1. In Eq. (5.8), set t = 1.
2. Differentiate Eq. (5.8) with respect to t and then set t = 0.
3. Differentiate Eq. (5.8) with respect to t and then set t = 1.
4. In Eq. (5.14), set t = u = v = 1.
5. Differentiate Eq. (5.14) with respect to t and then set t = u = v = 1.
6. Differentiate Eq. (5.14) with respect to u. Then set t = v = 1 and u = 0.
7. Differentiate Eq. (5.14) with respect to v. Then set t = u = 1 and v = 0.
The computer algebra quickly solved the linear system and then summed the generating
functions E1 and G1. The result can be seen in the following proposition.
S. Feretić: Polyominoes with nearly convex columns: A semi-directed model 123
Proposition 5.1. The area generating function for level two cheesy polyominoes is given
by
K =
q · (1− 6q + 11q2 − 6q3 − q4 − 3q6 + 5q7 + 4q8 − 3q9 − 3q10)
1− 9q + 27q2 − 31q3 + 8q4 + 4q5 − 2q6 + 16q7 − 5q8 − 16q9 − 2q10 + 5q11
.
Corollary 5.2. The number of n-celled level two cheesy polyominoes has the asymptotic
behaviour
[qn]K ∼ 0.121043 · 4.231836n.
Thus, the growth constant of level two cheesy polyominoes is 4.231836.
6 Level three cheesy polyominoes
The enumeration of level three cheesy polyominoes is similar to the enumeration of level
two cheesy polyominoes. However, there are still more cases than before. In the enu-
meration of level two cheesy polyominoes, the total number of cases was 9 (because we
partitioned the set U into 7 subsets and the set V into 2 subsets). At level three, the total
number of cases is 16. We deem it reasonable to skip those 16 cases and only state the final
result.
Proposition 6.1. The area generating function for level three cheesy polyominoes is given
by
L =
M
N
,
where
M = q · (1− 11q + 49q2 − 114q3 + 146q4 − 94q5 + 5q6 + 71q7 − 143q8
+ 176q9 − 154q10 + 100q11 + 24q12 − 121q13 + 90q14 − 61q15 + 19q16
+ 58q17 − 32q18 − 31q19 + 37q20 + 14q21 − 43q22 − 4q23 + 21q24
− q25 − 5q26)
and
N = 1− 14q + 80q2 − 243q3 + 423q4 − 413q5 + 174q6 + 106q7 − 350q8
+ 533q9 − 546q10 + 427q11 − 148q12 − 261q13 + 383q14 − 253q15
+ 158q16 + 57q17 − 181q18 + 10q19 + 115q20 − 49q21 − 96q22
+ 93q23 + 49q24 − 54q25 − 12q26 + 12q27 + q28.
Corollary 6.2. The number of n-celled level three cheesy polyominoes has the asymptotic
behaviour
[qn]L ∼ 0.108270 · 4.288631n.
124 Ars Math. Contemp. 5 (2012) 107–126
Column- Level 1 Level 2 Level 3
convex cheesy cheesy cheesy
polyo- polyo- polyo- polyo-
Area minoes minoes minoes minoes
1 1 1 1 1
2 3 3 3 3
3 11 11 11 11
4 42 43 43 43
5 162 173 174 174
6 626 705 718 719
7 2419 2889 2996 3012
8 9346 11867 12579 12727
9 36106 48795 52996 54067
10 139483 200723 223705 230464
11 538841 825845 945324 984477
12 2081612 3398081 3997185 4211222
Table 1: Here is how many polyominoes of a given type have 1, 2, . . . , 12 cells.
7 Taylor expansions and the limit value of the growth constants
To see how many polyominoes of a given type have 1, 2, 3, . . . cells, we expanded the area
generating functions into Taylor series. The results are shown in Table 1.
Next: How do the growth constants behave when level tends to infinity? Our database
is too small for making precise estimates. Anyway, we know that the growth constant of
column-convex polyominoes is 3.863, while the growth constants of level one, level two
and level three cheesy polyominoes are 4.115, 4.232 and 4.289, respectively. Computing
the first differences, we get the numbers 4.115 − 3.863 = 0.252, 4.232 − 4.115 = 0.117,
and 4.289− 4.232 = 0.057. Now, the sequence 0.252, 0.117, 0.057 is a little similar to a
geometric sequence with common ratio 12 . Hence, the limit value of the growth constants
of cheesy polyominoes might be about 4.289 + 0.057 = 4.346.
8 Conclusion
This paper is concerned with polyominoes on the hexagonal lattice. For every m ∈ N, we
have defined a set of polyominoes called level m cheesy polyominoes. A polyomino P is
a level m cheesy polyomino if the following holds:
1. P is a rightward-semi-directed polyomino,
2. every column of P has at most two connected components,
3. if a column of P has two connected components, then the gap between the compo-
nents consists of at most m cells.
Column-convex polyominoes are a subset of level one cheesy polyominoes and, for
every m ∈ N, level m cheesy polyominoes are a subset of level m+1 cheesy polyominoes.
For every m ∈ N, level m cheesy polyominoes have a rational area generating func-
tion. We have computed the area generating functions for levels one, two and three. At
S. Feretić: Polyominoes with nearly convex columns: A semi-directed model 125
those three levels, the number of n-celled cheesy polyominoes is asymptotically equal to
0.1441 . . .× 4.1149 . . .n, to 0.1210 . . .× 4.2318 . . .n, and to 0.1082 . . .× 4.2886 . . .n, re-
spectively. For comparison, the number of n-celled column-convex polyominoes is asymp-
totically equal to 0.1884 . . .× 3.8631 . . .n. The number of n-celled multi-directed animals
behaves asymptotically as constant×4.5878 . . .n [4]. (At present, multi-directed animals
are the largest exactly solved class of polyominoes. However, multi-directed animals are
not a superset of level one cheesy polyominoes.) The number of all n-celled polyominoes
behaves asymptotically as 0.2734...n × 5.1831 . . .
n [21].
9 Fresh news
The present paper is being published after many adventures. Meanwhile, this author has
generalized column-convex polyominoes in several other ways. The definition of cheesy
polyominoes (stated in Section 8) consists of three requirements. Now, in [9], the require-
ment that polyominoes are rightward-semi-directed is somehow relaxed. After this relax-
ation, the area generating functions are still rational but are somewhat less simple than in
the case of cheesy polyominoes. In [11, 12], the rightward-semi-directedness requirement
is completely removed. That is, the definition of the model is reduced to requirements No.
2 and 3 of the definition of cheesy polyominoes. For the model so defined, the area gener-
ating function is not a rational function, but a complicated q-series. It is also possible not to
require rightward-semi-directedness and, at the same time, allow two-component columns
to have gaps of all sizes. The only remaining requirement is then the second one, “ev-
ery column of P has at most two connected components”. The said requirement by itself
defines an unsolvable model, but that model can be made solvable by introducing a new re-
quirement. Indeed, in [11, 14], we forbid runs of two or more consecutive two-component
columns, and the area generating function is again a complicated, but computable, q-series.
All this does not mean, however, that cheesy polyominoes are no longer interesting.
In particular, it remains worthy of attention that level one cheesy polyominoes, which are
significantly more numerous than column-convex polyominoes, have an area generating
function which is even a bit simpler than the area generating function of column-convex
polyominoes.
Let us conclude with a few words about possibilities for future work. As far as I can
see, if requirement No. 2 is replaced by “every column of P has at most three connected
components”, the resulting model is still solvable, but if requirement No. 2 is replaced
by “every column of P has at most four connected components”, the resulting model is
unsolvable.
I think that, already at level one, cheesy polyominoes cannot be enumerated by perime-
ter. Namely, the perimeter generating function has zero radius of convergence, as can be
proved by adapting an argument given by Tony Guttmann in Section 9 of [11].
Cheesy polyominoes with square cells are an object which, in principle, can be stud-
ied. It ought to be mentioned, however, that hexagonal-celled cheesy polyominoes behave
somewhat better than their square-celled counterpart. Solving the square-celled level one
model requires almost as much effort as solving the hexagonal-celled level two model. The
same holds for the square-celled versions of the models of [9, 12].
126 Ars Math. Contemp. 5 (2012) 107–126
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