Enumeration of 4-stack polyominoes

Enumeration of 4-stack polyominoes


Simone Rinaldi1

(Joint work with A. Frosini, J.M. Fedou)

1University of Siena, Siena, Italy([email protected])

To appear on Theoretical Computer Science


4-stack polyominoes

L-convex polyominoes

A polyomino is L-convex if any two of its cells can beconnected by a path, internal to the polyomino, having atmost 1 change of direction.

A polyomino is Z-convex (or 2-convex) if any two of its cellscan be connected by a path, internal to the polyomino,having at most 2 changes of direction.


4-stack polyominoes

L-convex polyominoes

A polyomino is L-convex if any two of its cells can beconnected by a path, internal to the polyomino, having atmost 1 change of direction.A polyomino is Z-convex (or 2-convex) if any two of its cellscan be connected by a path, internal to the polyomino,having at most 2 changes of direction.


4-stack polyominoes

Centered polyominoes

A polyomino is stack if it contains at least two adjacentcorners of its minimal bounding rectangle.

A polyomino is centered if it contains at least one rowtouching both the left and the right side of its minimalbounding rectangle.Clearly, Ln ⊆ Cn ⊆ Zn.


4-stack polyominoes


A polyomino is stack if it contains at least two adjacentcorners of its minimal bounding rectangle.A polyomino is centered if it contains at least one rowtouching both the left and the right side of its minimalbounding rectangle.

Clearly, Ln ⊆ Cn ⊆ Zn.


4-stack polyominoes


A polyomino is stack if it contains at least two adjacentcorners of its minimal bounding rectangle.A polyomino is centered if it contains at least one rowtouching both the left and the right side of its minimalbounding rectangle.Clearly, Ln ⊆ Cn ⊆ Zn.


4-stack polyominoes

Enumeration results

the number ln of L-convex polyominoes withsemi-perimeter n + 2 satisfies:

ln = 4ln−1 − 2ln−2, n ≥ 3, with l0 = 1, l1 = 2, l2 = 7,

then ln ∼(

1+√

24

)(2 +√

2)n.

the number cn of centered convex polyominoes withsemi-perimeter n + 2 satisfies:

cn = 6cn−1 − 8cn−2, n ≥ 3, with c0 = 1, c1 = 2, c2 = 7,

then cn ∼ 384n.

the generating function of Z-convex polyominoes withrespect to the semi-perimeter is algebraic of degree 2, andthe number of Z-convex polyominoes havingsemi-perimeter n + 2 grows asymptotically as n

244n.


4-stack polyominoes

Comparison between classes of polyominoes

L-convex: rational, ln ∼ (2 +√

2)n.

Centered: rational, cn ∼ 4n.

4-stack polyominoes: an intermediate class.Marc Noy suggested that they are a rational class growingas n4n.Z-convex: algebraic, zn ∼ n4n.convex: algebraic, on ∼ n4n.


4-stack polyominoes



2)n.


4-stack polyominoes: an intermediate class.Marc Noy suggested that they are a rational class growingas n4n.

Z-convex: algebraic, zn ∼ n4n.

convex: algebraic, on ∼ n4n.


4-stack polyominoes



2)n.


4-stack polyominoes: an intermediate class.Marc Noy suggested that they are a rational class growingas n4n.

Z-convex: algebraic, zn ∼ n4n.convex: algebraic, on ∼ n4n.


4-stack polyominoes



2)n.


4-stack polyominoes: an intermediate class.Marc Noy suggested that they are a rational class growingas n4n.Z-convex: algebraic, zn ∼ n4n.convex: algebraic, on ∼ n4n.


4-stack polyominoes

4-stack polyominoes

A convex polyomino is 4-stack if it can be decomposed into arectangle (called central rectangle) supporting four (possiblyempty) stack polyominoes, one on each side of the rectangle(the up, right, down, and left stack of the polyomino).

down stack

central rectangle

up stack

left stack

right stack


4-stack polyominoes


In general, a 4-stack polyomino does not have a uniquedecomposition in terms of a central rectangle and 4 stacks.

We assume that the central rectangle is maximal, i.e. it isnot properly included in any other rectangle.Every 4-stack is a Z-convex polyomino, and every centeredpolyomino is a 4-stack.


4-stack polyominoes


In general, a 4-stack polyomino does not have a uniquedecomposition in terms of a central rectangle and 4 stacks.We assume that the central rectangle is maximal, i.e. it isnot properly included in any other rectangle.

Every 4-stack is a Z-convex polyomino, and every centeredpolyomino is a 4-stack.


4-stack polyominoes


In general, a 4-stack polyomino does not have a uniquedecomposition in terms of a central rectangle and 4 stacks.We assume that the central rectangle is maximal, i.e. it isnot properly included in any other rectangle.Every 4-stack is a Z-convex polyomino, and every centeredpolyomino is a 4-stack.


4-stack polyominoes


Since the decomposition of a 4-stack polyomino may notbe unique, we encounter ambiguity problems if we try tosolve enumeration problem by using a decompositionstrategy.

4-stack polyominoes resist standard enumerationtechniques applied for algebraic classes.We use a straightforward application of theinclusion/exclusion principle.


4-stack polyominoes


Since the decomposition of a 4-stack polyomino may notbe unique, we encounter ambiguity problems if we try tosolve enumeration problem by using a decompositionstrategy.4-stack polyominoes resist standard enumerationtechniques applied for algebraic classes.

We use a straightforward application of theinclusion/exclusion principle.


4-stack polyominoes


Since the decomposition of a 4-stack polyomino may notbe unique, we encounter ambiguity problems if we try tosolve enumeration problem by using a decompositionstrategy.4-stack polyominoes resist standard enumerationtechniques applied for algebraic classes.We use a straightforward application of theinclusion/exclusion principle.


4-stack polyominoes

k-colored 4-stack polyominoes

Let P be a 4-stack polyomino having at least kdecompositions. A k-colored 4-stack polyomino (briefly,k-colored polyomino) is any polyomino obtained from P bycoloring k different central rectangles using different colors.

To P̂ ∈ Sk , we associate a 4-stack polyomino im(P̂)obtained by forgetting the colors of the central rectangles:it is the image of P̂. Given a k -colored polyomino P̂, itsimage has at least k different decompositions, each onedetermined by one of the central rectangles.Clearly,S = im(S1) ⊃ im(S2) ⊃ . . . ⊃ im(Sk ) ⊃ im(Sk+1) ⊃ . . .


4-stack polyominoes


Let P be a 4-stack polyomino having at least kdecompositions. A k-colored 4-stack polyomino (briefly,k-colored polyomino) is any polyomino obtained from P bycoloring k different central rectangles using different colors.To P̂ ∈ Sk , we associate a 4-stack polyomino im(P̂)obtained by forgetting the colors of the central rectangles:it is the image of P̂. Given a k -colored polyomino P̂, itsimage has at least k different decompositions, each onedetermined by one of the central rectangles.

Clearly,S = im(S1) ⊃ im(S2) ⊃ . . . ⊃ im(Sk ) ⊃ im(Sk+1) ⊃ . . .


4-stack polyominoes


Let P be a 4-stack polyomino having at least kdecompositions. A k-colored 4-stack polyomino (briefly,k-colored polyomino) is any polyomino obtained from P bycoloring k different central rectangles using different colors.To P̂ ∈ Sk , we associate a 4-stack polyomino im(P̂)obtained by forgetting the colors of the central rectangles:it is the image of P̂. Given a k -colored polyomino P̂, itsimage has at least k different decompositions, each onedetermined by one of the central rectangles.Clearly,S = im(S1) ⊃ im(S2) ⊃ . . . ⊃ im(Sk ) ⊃ im(Sk+1) ⊃ . . .


4-stack polyominoes


If P is a 4-stack polyomino having exactly h decompositions,then, for all k ≤ h, there are exactly q =

(hk

)k -colored

polyominoes P̂1, . . . , P̂q in Sk such thatim(P̂1) = . . . = im(P̂q) = P, i.e. having P as their image.Example: 4-stack polyominoes with exactly 3 decompositions.

(f) (g) (h)

(a) (b) (c) (d)

(e)


4-stack polyominoes


Our strategy is to obtain enumeration of 4-stack polyominoesby enumerating k -colored polyominoes. Thus, let Sn (resp. Sk

n )denote the class of 4-stack polyominoes (resp. k -coloredpolyominoes) of semi-perimeter equal to n. Using a classicalinclusion/exclusion argument, we have the following:

Proposition

For any n ≥ 2, we have

|Sn| =∑i≥1

(−1)i+1∣∣∣Si

n

∣∣∣ .


4-stack polyominoes


Using standard analytic techniques we can obtain thegenerating function of centered polyominoes according to thenumber of rows and columns:

C(x , y) =xy(1− y − 2x − xy + x2)(1− y)

(1− x − y)(1− 2y − 2x − 2xy + x2 + y2).

(a)

(b)


4-stack polyominoes

Enumeration of 1-colored polyominoes polyominoes

Let P̂ be a 1-colored polyomino having h rows and w columns.Moreover, let w ′ and h′ be the basis and the height of thecolored maximal rectangle of P̂. As it is shown, P̂ can beuniquely decomposed into two centered polyominoes: the firstone is vertically centered (the central rectangle has null width)with dimensions (w − w ′,h′); the second one is horizontallycentered (the central rectangle has null height) and dimensions(w ′,h − h′).

h’

w’

h−h’h

w

w’

h’

w−w’


4-stack polyominoes

Enumeration of 1-colored polyominoes polyominoes

The generating function of 1-colored polyominoes according tothe width and the height can be obtained from the previouscomputation, and it is equal to

S1(x , y) = C>(x , y) · C>(y , x)

where C>(x , y) are centered polyominoes where the height ofthe central rectangle is null.


4-stack polyominoes

Bi-centered paths

Let us consider the points A = (−a,0), B = (0,b), C = (c,0),and D = (0,−d), with a,b, c,d ≥ 0. A closed path π, startingfrom A and running clockwise, is a bi-centered path if it can bedecomposed into the following subpaths:

1 the path π1, running from A to B using N and E unit steps;

2 the path π2, running from B to C using S and E unit steps;3 the path π3, running from C to D using S and W unit steps;4 the path π4, running from D to A using N and W unit steps.

��

��

(b)

A

B

C

D

(a)

A

B

C

D


4-stack polyominoes

Bi-centered paths


1 the path π1, running from A to B using N and E unit steps;2 the path π2, running from B to C using S and E unit steps;

3 the path π3, running from C to D using S and W unit steps;4 the path π4, running from D to A using N and W unit steps.

��

��

(b)

A

B

C

D

(a)

A

B

C

D


4-stack polyominoes

Bi-centered paths


1 the path π1, running from A to B using N and E unit steps;2 the path π2, running from B to C using S and E unit steps;3 the path π3, running from C to D using S and W unit steps;

4 the path π4, running from D to A using N and W unit steps.

��

��

(b)

A

B

C

D

(a)

A

B

C

D


4-stack polyominoes

Bi-centered paths


1 the path π1, running from A to B using N and E unit steps;2 the path π2, running from B to C using S and E unit steps;3 the path π3, running from C to D using S and W unit steps;4 the path π4, running from D to A using N and W unit steps.

��

��

(b)

A

B

C

D

(a)

A

B

C

D


4-stack polyominoes

Bi-centered paths

We say that a bi-centered path π is improper when there is arectangle inside the polygon that touches the boundingrectangle of π in two opposite sides.

��

��

(b)

A

B

C

D

(a)

A

B

C

D


4-stack polyominoes

Enumeration of k-colored polyominoes

Proposition

For any k > 1, a k-colored polyomino with dimensions (w ,h)con be uniquely decomposed as follows:- a 1-colored polyomino with dimensions (w ′,h′),- k − 1 proper bi-centered paths with dimensions(w ′1,h

′1), . . . (w

′k−1,h

′k−1),

with w ′ + w ′1 + . . .+ w ′k−1 = w, h′ + h′1 + . . .+ h′k−1 = h.


4-stack polyominoes


U

W =WZ

WZ

V

VU

U

V

Z

3

2

3

12

3

21

11

kb

1h

2

3

A 3-colored polyomino uniquely decomposed into a 1-colored polyomino and two proper bi-centered paths.


4-stack polyominoes


The generating function for the number of bi-centered pathspn,m is

B(x , y) =∑

n,m≥0

pn,m xnym =1

(1− x − y)√

1− 2 y − 2 x + x2 − 2 xy + y2

The generating function B(x , y) of proper bicentered paths, isobtained by observing that any bicentered path π′ is either thepath having null width and height, or it can be obtained from aproper bicentered path π by adding appropriate sequences ofhorizontal or vertical steps. Then B(x , y) and B(x , y) arerelated by the formula

B(x , y) = 1 +B(x , y)

(1− x)2(1− y)2 ,


4-stack polyominoes


Proposition

For any k > 1, the generating function Sk (x , y) of k-coloredpolyominoes, according to width and height, is given by

Sk (x , y) = S1(x , y)Bk−1

(x , y) .

We observe that, while the generating function of 1-coloredpolyominoes is rational, the generating functions of k -coloredpolyominoes, with k > 1, are all algebraic ones.


4-stack polyominoes


Theorem

The generating function S(x , y) of 4-stack polyominoes,according to width and height, is given by

S(x , y) =S1(x , y)

1 + B(x , y)

=xy(1− 2 x − y − xy + x2) (1− x − 2 y − xy + y2)

(1− x − y)(1− 2 x − 2 y − 2 xy + x2 + y2

)3/2


4-stack polyominoes

The generating function of 4-stack polyominoes accordingto sp is algebraic,

S(x) =x2(1− 3x)2

(1− 2x)(1− 4x)32

sequence 1,2,7,28,116,484,2018,8392,34802, . . .

The number of 4-stack polyominoes with semi-perimetern + 2 is given by

sn =

(2nn

)+

n−1∑h=0

(2hh

)(2n−h−1

2

)n ≥ 0 .

The sequence sn asymptotically grows like√

n4n, precisely

sn ∼1

64√π·√

n · 4n ,

which is just below the term n4n conjectured by Marc Noy.


4-stack polyominoes

The generating function of 4-stack polyominoes accordingto sp is algebraic,

S(x) =x2(1− 3x)2

(1− 2x)(1− 4x)32

sequence 1,2,7,28,116,484,2018,8392,34802, . . .The number of 4-stack polyominoes with semi-perimetern + 2 is given by

sn =

(2nn

)+

n−1∑h=0

(2hh

)(2n−h−1

2

)n ≥ 0 .

The sequence sn asymptotically grows like√

n4n, precisely

sn ∼1

64√π·√

n · 4n ,

which is just below the term n4n conjectured by Marc Noy.


4-stack polyominoes

Finally, the generating function for the number of bi-centeredpolyominoes according to rows and columns is given by

x+ y+ B(x , y)

and the generating function according to sp is

Bi(x) =

(1− 2 x + x2) x2

(1− 2 x)√

1− 4 x


4-stack polyominoes

Among these classes, we have the following inclusions:

L-convex ⊂ bi-centered ⊂ centered ⊂ 4-stack ⊂ Z-convex ⊂ convex .

In summary, comparing the natures and the asymptoticbehaviors of these classes we have following:

Class of polyominoes Type of g.f. Asymptotic growth

bi-centered algebraic 9128√π· 4n√

n

centered rational 38 · 4

n

4-stack algebraic 164√π·√

n · 4n

Z-convex algebraic n24 · 4

n


4-stack polyominoes

References

G. Castiglione, A. Frosini, E. Munarini, A. Restivo, S.Rinaldi, Combinatorial aspects of L-convex polyominoes,European J. Combin. 28 (2007) 1724-1741.G. Castiglione, A. Frosini, A. Restivo, S. Rinaldi,Enumeration of L-convex polyominoes by rows andcolumns, Theor. Comp. Sci., 347 (2005) 336–352.E. Duchi, S. Rinaldi, G. Schaeffer, The number of Z-convexpolyominoes, Adv. Appl. Math. 4 (2008) 54-72.M. Noy, Some comments on 2-convex polyominoes,Personal communication.


4-stack polyominoes

References

G. Castiglione, A. Frosini, E. Munarini, A. Restivo, S.Rinaldi, Combinatorial aspects of L-convex polyominoes,European J. Combin. 28 (2007) 1724-1741.

G. Castiglione, A. Frosini, A. Restivo, S. Rinaldi,Enumeration of L-convex polyominoes by rows andcolumns, Theor. Comp. Sci., 347 (2005) 336–352.E. Duchi, S. Rinaldi, G. Schaeffer, The number of Z-convexpolyominoes, Adv. Appl. Math. 4 (2008) 54-72.M. Noy, Some comments on 2-convex polyominoes,Personal communication.


4-stack polyominoes

References

G. Castiglione, A. Frosini, E. Munarini, A. Restivo, S.Rinaldi, Combinatorial aspects of L-convex polyominoes,European J. Combin. 28 (2007) 1724-1741.G. Castiglione, A. Frosini, A. Restivo, S. Rinaldi,Enumeration of L-convex polyominoes by rows andcolumns, Theor. Comp. Sci., 347 (2005) 336–352.

E. Duchi, S. Rinaldi, G. Schaeffer, The number of Z-convexpolyominoes, Adv. Appl. Math. 4 (2008) 54-72.M. Noy, Some comments on 2-convex polyominoes,Personal communication.


4-stack polyominoes

References

G. Castiglione, A. Frosini, E. Munarini, A. Restivo, S.Rinaldi, Combinatorial aspects of L-convex polyominoes,European J. Combin. 28 (2007) 1724-1741.G. Castiglione, A. Frosini, A. Restivo, S. Rinaldi,Enumeration of L-convex polyominoes by rows andcolumns, Theor. Comp. Sci., 347 (2005) 336–352.E. Duchi, S. Rinaldi, G. Schaeffer, The number of Z-convexpolyominoes, Adv. Appl. Math. 4 (2008) 54-72.

M. Noy, Some comments on 2-convex polyominoes,Personal communication.


4-stack polyominoes

References

G. Castiglione, A. Frosini, E. Munarini, A. Restivo, S.Rinaldi, Combinatorial aspects of L-convex polyominoes,European J. Combin. 28 (2007) 1724-1741.G. Castiglione, A. Frosini, A. Restivo, S. Rinaldi,Enumeration of L-convex polyominoes by rows andcolumns, Theor. Comp. Sci., 347 (2005) 336–352.E. Duchi, S. Rinaldi, G. Schaeffer, The number of Z-convexpolyominoes, Adv. Appl. Math. 4 (2008) 54-72.M. Noy, Some comments on 2-convex polyominoes,Personal communication.

Enumeration of 4-stack polyominoes

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