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1 Introduction
Polyominoes are objects sprung from recreative mathematics and from different
domains in physics (such as Ising's model;
its generalisation, Pott's model; directed percolation and branched
polymer problems) [14, 20, 21, 25].
Two great classes of problems relative to polyominoes are
tiling problems;
enumeration problems.
David Klarner began to study polyomino tilings in 1965.
There are still open questions in this
field [12, 15, 16, 17], however
several (un)decidability results are known [1, 2].
What is more, aperiodic tilings are today a new spring of inspiration in
noncommutative geometry [8].
In the remainder, we only consider enumeration problems.
Exact asymptotics of polyominoes on a square lattice is still unknown.
Accurate results are then limited to special
families of polyominoes, for which we know a generative grammar. We are
therefore brought back to the study of a functional equation which
defines the generating function. Nevertheless, obtaining of a closed
form (i.e., an explicit solution) or even any form of solution
often remains difficult. We will show several methods to obtain them.
2 Definitions
A polyomino is a connected set on a lattice.
A polyomino is said to be convex if it is both columnconvex and rowconvex.
A polyomino is said to be directed if, for each couple of points of the
polyomino, there exists
a path only made of North and West steps which links this two points.
One can find in previous summaries [4, 13] how to obtain
functional equations satisfied by the generating functions
(most of the methods are tricky decompositions [5] of polyominoes
into very regular smaller pieces, such as ``strata'' or ``waspwaist''
decompositions). For results in dimension greater than 2,
see [3, 6].
3 Differential Equation Method
Enumeration of convex polyominoes with perimeter 2n on the honeycomb
(or ``hexagonal'') lattice can be solved with this method.
Let P_{n} the number of such polyominoes with perimeter 2n+6,
Enting [10] gives the following result.
The generating function P(x)=å_{n=0}^{¥}p_{n}x^{n} satisfies
the differential equation
Let us mention that the package GFUN in Maple is able to make
such translations (recurrences, differential equations, algebraic equations,
closed forms), see [23].
4 Temperley's Method
We are going to illustrate Temperley's method [24] with the
enumeration of column convex polyominoes (on a square lattice)
with respect to perimeter [7].
The generating function
and g_{1}, g_{2}, g_{3}, g_{4}, the ``initial conditions'', are known.
If we ``guess'' that g_{r} has the shape l^{r} (or is a linear combination of such
monomials),
we can obtain l by solving the fourth degree equation associated
to the recurrence formula, and we find, as the equation easily splits:
So solving the two second degree equations gives four values (closed
forms)
l_{1}, l_{2}, l_{3}, l_{4}, two of which are O(1) at 0.
We then have to find the A_{j} such that g_{r}=å_{j=1}^{4}A_{j} l_{j}^{r}.
But g_{r}=O(y^{2r+2}) at 0, so A_{i}=0 if l_{i}=O(1).
There are still two coefficients to determine, say A_{2} and A_{4}.
They can be found by solving a system involving
g_{1}, g_{2}, l_{2}, l_{4}, A_{2}, A_{4} and one finally obtains a closed form
G(y)=
A_{2} l_{2}
1l_{2}
+
A_{4} l_{4}
1l_{4}
.
A very similar method is applied for
unidirectionalconvex polygons on the honeycomb lattice in [19].
5 Kernel Method
In his talk, Dominique GouyouBeauchamps has also presented an
exploitation of the ``kernel method'' for the enumeration of
parallelogram polyominoes
with respect to horizontal and vertical halfperimeter, area and first column
height, respectively marked by x, y, q, s.
Remember that the generating function P with respect to
horizontal and vertical halfperimeter is easy to obtain:
The waspwaist decomposition directly leads to P=xy+xP+yP+P^{2} so
P(x, y)=
1xy(12x2y+x^{2}+y^{2}2xy)^{1/2}
2
.
The full generating function P(x, y, q, s) satisfies a more
intricate equation (obtained by a strata decomposition), namely
P(x, y, q, s)=
xysq
1ysq
+
xsq
(1sq)(1ysq)
P(x, y, q, 1)
xsq
(1sq)(1ysq)
P(x, y, q, sq).
When q=1, this can be rewritten
(1(1xy)s+y^{2}) P(x, y, 1, s) =xsP(x, y, 1, 1)+xys(1s).
It is typically the type of equation on which the kernel method applies.
This method belongs to mathematical folklore (see [18],
exercise 2.2.1.4 for an early example).
It works as follows: If one cancels the kernel (1(1xy)s+y^{2}),
i.e., one finds s_{0} such that (1(1xy)s_{0}+y^{2})=0,
then one gets 0=xs_{0}P(x, y, 1, 1)+xys_{0}(1s_{0}), from which follows
a closed form for P(x, y, 1, 1)
and finally one obtains a closed form for P(x, y, 1, s), viz.,
P(x, y, 1, s) =
xs
æ ç ç è
1xy(12x2y2xy+x^{2}+y^{2})^{1/2}
2
ö ÷ ÷ ø
+xys(1s)
1(1xy)s+y^{2}
.
6 Physicists' Guesses
We have already mentioned that polyominoes are present in physical
problems and in fact the first people who found interesting results
on this subject where physicists. They sometimes base their works
on empirical results. For example, in [9], the authors are doing as if
N_{r}^{s}=N_{r1}^{s1}+N_{r}^{s1}+N_{r+1}^{s1}
(N_{r}^{s} is the number of
directed animals of size s with a ``compact source'' of size r)
was a recurrence formula satisfied by the N_{r}^{s}
although it is only empirically verified for the first values.
Nevertheless, they go on and find that
N_{r}^{s}=
1
2p
ó õ
2 p
0
(1+e^{it}) e^{irt} (1+2 cos t)^{s1}dt
and in particular
N_{1}^{s}=(s1)!
ë s/2û
å
q=0
sq
q!^{2} (s2q)!
.
Another example of a typical physicist's method is [14] (enumeration of
directed animals on a strip of width k);
they consider a transfer matrix as an operator acting on a spin space
and are drawing their inspiration from standard techniques on integrable systems.
When k tends to infinity, they obtain:
a_{n}=
å
0£ i £ n
æ è
n1
i
ö ø
æ è
i
ë i/2 û
ö ø
and thus
å
n³ o
a_{n}t^{n} =
1
2
æ ç ç è
(
1+t
13t
)^{1/2} 1
ö ÷ ÷ ø
.
Analysis of singularities gives
a_{n} ~ 3^{n}n^{1/2}.
7 Matricial and Continued Fraction Method
We will show on a simple example (Dyck paths) how this method works.
Let
d_{h}(x)=
å
l³ 0
a_{h, l}x^{l}
the ordinary generating function of Dyck paths which end at height h.
A path of length n which ends at height h is
either a path of length n1 which ends at height h1 followed by
a NE step,
or a path of length n1 which ends at height h+1 followed by
a SE step. Thus one obtains the following infinite system
ì ï ï ï ï í ï ï ï ï î
d_{0}(x)=1+xd_{1}(x)
d_{1}(x)=xd_{0}(x)+xd_{2}(x)
d_{2}(x)=xd_{1}(x)+xd_{3}(x)
· · ·
d_{h}(x)=xd_{h1}(x)+xd_{h+1}(x)
· · ·
which can be written as
æ ç ç ç ç ç ç è
1
x
0
0
...
x
1
x
0
...
0
x
1
x
...
0
0
x
1
...
· · ·
· · ·
· · ·
· · ·
· · ·
ö ÷ ÷ ÷ ÷ ÷ ÷ ø
æ ç ç ç ç ç ç è
d_{0}(x)
d_{1}(x)
d_{2}(x)
d_{3}(x)
· · ·
ö ÷ ÷ ÷ ÷ ÷ ÷ ø
=
æ ç ç ç ç ç ç è
1
0
0
0
· · ·
ö ÷ ÷ ÷ ÷ ÷ ÷ ø
.
With an analog of Cramer's formula for infinite matrices, one has
d_{0}(x)=
det
æ ç ç ç ç ç ç è
1
x
0
0
...
0
1
x
0
...
0
x
1
x
...
0
0
x
1
...
· · ·
· · ·
· · ·
· · ·
· · ·
ö ÷ ÷ ÷ ÷ ÷ ÷ ø
det
æ ç ç ç ç ç ç è
1
x
0
0
...
x
1
x
0
...
0
x
1
x
...
0
0
x
1
...
· · ·
· · ·
· · ·
· · ·
· · ·
ö ÷ ÷ ÷ ÷ ÷ ÷ ø
=
lim
k® ¥
det
(
)
k× k
det
(
)
k× k
=
lim
k®
¥
P_{k}(x)
Q_{k}(x)
where ()_{k × k}
stands for the k× k truncated associated matrices.
The special structure of these matrices gives the recurrence
ì í î
P_{k}(x)=Q_{k1}(x)=P_{k1}(x)x^{2}P_{k2}(x)
with P_{1}(x)=1,
Q_{k}(x)=Q_{k1}(x)x^{2}Q_{k2}(x)
with Q_{1}(x)=1 and Q_{2}(x)=1x^{2}.
from which follows
P_{k}(x)
Q_{k}(x)
=
Q_{k1}(x)
Q_{k}(x)
=
Q_{k1}(x)
Q_{k1}(x)x^{2}Q_{k2}(x)
=
1
1x^{2}
Q_{k2}(x)
Q_{k1}(x)
=
1
1x^{2}
P_{k1}(x)
Q_{k1}(x)
and then
d_{0}(x)=
lim
k® ¥
P_{k}(x)
Q_{k}(x)
=
1
1
x^{2}
1
x^{2}
· · ·
hence
d_{0}(x)=
1
1x^{2}d_{0}(x)
i.e., d_{0}(x)=
1(14x^{2})^{1/2}
2 x^{2}
.
In fact the continued fraction is a special case
of a much more general result that we will express in the next section.
8 Multicontinued Fractions Theorem
We will need the following notations. Let
(l_{l, k})_{0£ k £ l} be a family of elements of a
commutative field and let (P_{k})_{k³ 0} be a family of monic polynomials
which satisfy a recurrence relation:
P_{k+1}(x)=xP_{k}(x)
k
å
i=0
l_{k, ki}P_{ki}(x).
One then defines a multicontinued fraction by
L(l, t)=
1
1l_{0, 0}t

¥
å
p=1
l_{p, 0}t^{p+1}
p
Õ
i=1
1
1l_{i, i}t

¥
å
q=1
l_{q+i, 0}t^{q+1}
q
Õ
i=1
1
· · ·
.
Let d be the operator defined by d(l_{k, l})=l_{k+1, l+1}.
We note P^{*} the reciprocal polynomial of P:
P^{*}(x):= x
deg(P)
P(
1
x
).
Theorem 1 [Roblet, Viennot]
If one sets l_{i, j}:=0 in L(l, t) for i³ k+1 and j£ i,
one gets a rational fraction L_{k}(t), it is the kth convergent of the
multicontinued fraction L(l, t) and we have
L_{k}(t)=
d P_{k}^{*}(t)
P_{k+1}^{*} (t)
and the following approximation near t=0 holds
L(l, t)=L_{k}(t)+O(t^{k+1}).
For a deeper understanding of links between continued fractions
and combinatorics, see [11, 22].
The multicontinued fraction method allows to find the generating functions of
diagonally convex directed, diagonally convex, parallelogram,
vertically convex directed, vertically convex polyominoes
and remains to be exploited to obtain generating functions of other
classes of polyominoes or directed animals.
You are now ready to try the different kinds of methods presented here
on your favourite class of polyominoes or even on other classes
of combinatorial objects!
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