Periodic Parallelogram Polyominoes
P. Laborde-Zubieta
joint work with A. Boussicault
LaBRI - Université de Bordeaux
GASCOM 2016 - 04/06/2016
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Outline
Definitions
Decomposition of PPPs in ordered trees
2 / 17
Parallelogram Polyominoes
A parallelogram polyomino is a set of edge connected cells, whose
boundary can be decomposed in two paths : a lower path and an
upper path. They are made of north and east steps, and they meet
only at their starting and ending point.
3 / 17
Periodic Parallelogram Polyominoes
A periodic parallelogram polyomino (PPP) is a parallelogram
polyomino with a marked cell in the rightmost column.
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Periodic Parallelogram Polyominoes
A periodic parallelogram polyomino (PPP) is a parallelogram
polyomino with a marked cell in the rightmost column.
The marking corresponds to the location where we “glue” the
leftmost and the rightmost columns. The number of cells above
the marked cell, included, is lower or equal to the number of cells
in the leftmost column.
4 / 17
Periodic Parallelogram Polyominoes
A periodic parallelogram polyomino (PPP) is a parallelogram
polyomino with a marked cell in the rightmost column.
The marking corresponds to the location where we “glue” the
leftmost and the rightmost columns. The number of cells above
the marked cell, included, is lower or equal to the number of cells
in the leftmost column.
4 / 17
Periodic Parallelogram Polyominoes
A periodic parallelogram polyomino (PPP) is a parallelogram
polyomino with a marked cell in the rightmost column.
The marking corresponds to the location where we “glue” the
leftmost and the rightmost columns. The number of cells above
the marked cell, included, is lower or equal to the number of cells
in the leftmost column.
4 / 17
Height, width, semi-perimeter
I
Height : the number of rows below the marked row
I
Width : the number of columns
I
Semi-perimeter : the height + the width
1
2
4
5 4 3
5
6
1
8 7
2
3
4
5
5 / 17
Rotation and strips
Making a rotation to a PPP means, putting the leftmost column in
the rightmost position and updating the marking.
rotation
6 / 17
Rotation and strips
Making a rotation to a PPP means, putting the leftmost column in
the rightmost position and updating the marking.
rotation
The rotation induces a partitioning of PPPs in equivalent classes,
we call them strips. Since the rotation does not modify the
semi-perimeter, we call semi-perimeter of a strip, the
semi-perimeter of one of its PPP.
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Plan
Definitions
Decomposition of PPPs in ordered trees
7 / 17
Underlying graph of a PPP
The blue and red labels are there to help the understanding, they
are not necessary.
1
4
2
5 4 3
6
1
8 7
2
3
4
5
The sons of a vertex are ordered according to their order in the
PPP.
The number of vertices of the graph is equal to the semi-perimeter
of the PPP.
The graph is bipartite, hence each cycle is of even size.
All the PPP of a same Strip have the same underlying graph.
8 / 17
Underlying graph of a PPP
The blue and red labels are there to help the understanding, they
are not necessary.
1
4
2
5 4 3
6
1
8 7
2
3
4
5
The sons of a vertex are ordered according to their order in the
PPP.
The number of vertices of the graph is equal to the semi-perimeter
of the PPP.
The graph is bipartite, hence each cycle is of even size.
All the PPP of a same Strip have the same underlying graph.
8 / 17
Underlying graph of a PPP
The blue and red labels are there to help the understanding, they
are not necessary.
1
2
4
5 4 3
5
6
1
8 7
2
3
4
5
The sons of a vertex are ordered according to their order in the
PPP.
The number of vertices of the graph is equal to the semi-perimeter
of the PPP.
The graph is bipartite, hence each cycle is of even size.
All the PPP of a same Strip have the same underlying graph.
8 / 17
Underlying graph of a PPP
The blue and red labels are there to help the understanding, they
are not necessary.
1
1
2
1
4
5 4 3
5
6
5
1
8 7
2
3
4
5
2
4
3
8
4
2
3
6
5
7
The sons of a vertex are ordered according to their order in the
PPP.
The number of vertices of the graph is equal to the semi-perimeter
of the PPP.
The graph is bipartite, hence each cycle is of even size.
All the PPP of a same Strip have the same underlying graph.
8 / 17
Underlying graph of a PPP
The blue and red labels are there to help the understanding, they
are not necessary.
1
1
2
1
4
5 4 3
5
6
5
1
8 7
2
3
4
5
2
4
3
8
4
2
3
6
5
7
The sons of a vertex are ordered according to their order in the
PPP.
The number of vertices of the graph is equal to the semi-perimeter
of the PPP.
The graph is bipartite, hence each cycle is of even size.
All the PPP of a same Strip have the same underlying graph.
8 / 17
Underlying graph of a PPP
The blue and red labels are there to help the understanding, they
are not necessary.
1
1
2
1
4
5 4 3
5
6
5
1
8 7
2
3
4
5
2
4
3
8
4
2
3
6
5
7
The sons of a vertex are ordered according to their order in the
PPP.
The number of vertices of the graph is equal to the semi-perimeter
of the PPP.
The graph is bipartite, hence each cycle is of even size.
All the PPP of a same Strip have the same underlying graph.
8 / 17
Underlying graph of a PPP
The blue and red labels are there to help the understanding, they
are not necessary.
1
1
2
1
4
5 4 3
5
6
5
1
8 7
2
3
4
5
2
4
3
8
4
2
3
6
5
7
The sons of a vertex are ordered according to their order in the
PPP.
The number of vertices of the graph is equal to the semi-perimeter
of the PPP.
The graph is bipartite, hence each cycle is of even size.
All the PPP of a same Strip have the same underlying graph.
8 / 17
Underlying graph of a PPP
The blue and red labels are there to help the understanding, they
are not necessary.
1
1
2
1
4
5 4 3
5
6
5
1
8 7
2
3
4
5
2
4
3
8
4
2
3
6
5
7
The sons of a vertex are ordered according to their order in the
PPP.
The number of vertices of the graph is equal to the semi-perimeter
of the PPP.
The graph is bipartite, hence each cycle is of even size.
All the PPP of a same Strip have the same underlying graph.
8 / 17
Description of graphs
Since the number of vertices is equal to the number of edges in
each connected component, by the following formula,
cyclomatic number − 1 = #edges − #vertices,
the cyclomatic number of each connected component is equal to 1.
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Description of graphs
Since the number of vertices is equal to the number of edges in
each connected component, by the following formula,
cyclomatic number − 1 = #edges − #vertices,
the cyclomatic number of each connected component is equal to 1.
Moreover, there is at least one cycle per connected component.
9 / 17
Description of graphs
Since the number of vertices is equal to the number of edges in
each connected component, by the following formula,
cyclomatic number − 1 = #edges − #vertices,
the cyclomatic number of each connected component is equal to 1.
Moreover, there is at least one cycle per connected component.
Hence, each connected component is a cycle with two ordered
trees attached to each vertex of the cycle, one “inside” and one
“outside”.
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Pruning
Pruning a PPP consists in removing recursively the rows and the
columns corresponding to leaves in the underlying grah.
1
1
2
1
4
5 4 3
5
6
5
1
8 7
2
3
4
5
2
4
3
8
4
2
3
6
5
7
How do we reconstruct the leaf ?
We have two possibilities.
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Pruning
Pruning a PPP consists in removing recursively the rows and the
columns corresponding to leaves in the underlying grah.
1
1
2
1
4
5 4 3
5
6
5
1
8 7
2
3
4
5
2
4
3
8
4
2
3
6
5
7
How do we reconstruct the leaf ?
We have two possibilities.
10 / 17
Pruning
Pruning a PPP consists in removing recursively the rows and the
columns corresponding to leaves in the underlying grah.
1
1
2
1
4
5 4 3
5
6
5
1
8
2
3
4
5
2
4
3
8
4
2
3
6
5
7
How do we reconstruct the leaf ?
We have two possibilities.
10 / 17
Pruning
Pruning a PPP consists in removing recursively the rows and the
columns corresponding to leaves in the underlying grah.
1
1
2
1
4
5 4 3
5
6
5
1
8
2
3
4
5
2
4
3
8
4
2
3
6
5
7
How do we reconstruct the leaf ?
We have two possibilities.
10 / 17
Pruning
Pruning a PPP consists in removing recursively the rows and the
columns corresponding to leaves in the underlying grah.
1
1
2
1
4
5 4
5
6
5
1
8
2
3
4
5
2
4
3
8
4
2
3
6
5
7
How do we reconstruct the leaf ?
We have two possibilities.
10 / 17
Pruning
Pruning a PPP consists in removing recursively the rows and the
columns corresponding to leaves in the underlying grah.
1
1
2
1
4
5 4
5
6
5
1
8
2
3
4
5
2
4
3
8
4
2
3
6
5
7
How do we reconstruct the leaf ?
We have two possibilities.
10 / 17
Pruning
Pruning a PPP consists in removing recursively the rows and the
columns corresponding to leaves in the underlying grah.
1
1
2
1
4
5 4
5
6
5
1
2
3
4
5
2
4
3
8
4
2
3
6
5
7
How do we reconstruct the leaf ?
We have two possibilities.
10 / 17
Pruning
Pruning a PPP consists in removing recursively the rows and the
columns corresponding to leaves in the underlying grah.
How do we reconstruct the leaf ?
We have two possibilities.
10 / 17
Pruning
Pruning a PPP consists in removing recursively the rows and the
columns corresponding to leaves in the underlying grah.
How do we reconstruct the leaf ?
We have two possibilities.
10 / 17
Pruning
Pruning a PPP consists in removing recursively the rows and the
columns corresponding to leaves in the underlying grah.
How do we reconstruct the leaf ?
We have two possibilities.
10 / 17
Pruning
Pruning a PPP consists in removing recursively the rows and the
columns corresponding to leaves in the underlying grah.
How do we reconstruct the leaf ?
We have two possibilities.
10 / 17
Pruning
Pruning a PPP consists in removing recursively the rows and the
columns corresponding to leaves in the underlying grah.
To be able to reconstruct back, we need to mark the vertex
corresponding to the leftmost column.
10 / 17
Pruning
Pruning a PPP consists in removing recursively the rows and the
columns corresponding to leaves in the underlying grah.
1
1
2
1
4
5 4
5
6
5
1
2
3
4
5
2
4
3
8
2
3
4
6
5
7
To be able to reconstruct back, we need to mark the vertex
corresponding to the leftmost column.
10 / 17
Pruning
Pruning a PPP consists in removing recursively the rows and the
columns corresponding to leaves in the underlying grah.
1
1
1
2
4
5 4
5
5
6
1
4
4
3
2
5
2
8
2
3
4
6
5
7
To be able to reconstruct back, we need to mark the vertex
corresponding to the leftmost column.
10 / 17
Pruning
Pruning a PPP consists in removing recursively the rows and the
columns corresponding to leaves in the underlying grah.
1
1
1
2
4
5 4
5
5
6
1
4
4
3
2
5
2
8
2
3
4
6
5
7
To be able to reconstruct back, we need to mark the vertex
corresponding to the leftmost column.
10 / 17
Pruning
Pruning a PPP consists in removing recursively the rows and the
columns corresponding to leaves in the underlying grah.
1
1
1
2
4
4
5
5
6
1
4
4
3
2
5
2
8
2
3
4
6
5
7
To be able to reconstruct back, we need to mark the vertex
corresponding to the leftmost column.
10 / 17
Trunk PPP + intrinsic thickness
After pruning a PPP, we will always get a PPP of the following
shape. Such a PPP is called a trunk PPP.
We define the intrinsic thickness of a trunk PPP to be the height
of one of its column minus 1. A trunk PPP is completely
determined by its number of columns and its intrinsic thickness.
The intrinsic thickness of a general PPP, is the intrinsic thickness
of the trunk PPP obtained after pruning it.
11 / 17
Trunk PPP + intrinsic thickness
After pruning a PPP, we will always get a PPP of the following
shape. Such a PPP is called a trunk PPP.
We define the intrinsic thickness of a trunk PPP to be the height
of one of its column minus 1. A trunk PPP is completely
determined by its number of columns and its intrinsic thickness.
The intrinsic thickness of a general PPP, is the intrinsic thickness
of the trunk PPP obtained after pruning it.
11 / 17
Trunk PPP + intrinsic thickness
After pruning a PPP, we will always get a PPP of the following
shape. Such a PPP is called a trunk PPP.
We define the intrinsic thickness of a trunk PPP to be the height
of one of its column minus 1. A trunk PPP is completely
determined by its number of columns and its intrinsic thickness.
The intrinsic thickness of a general PPP, is the intrinsic thickness
of the trunk PPP obtained after pruning it.
11 / 17
Trunk PPP + intrinsic thickness
After pruning a PPP, we will always get a PPP of the following
shape. Such a PPP is called a trunk PPP.
We define the intrinsic thickness of a trunk PPP to be the height
of one of its column minus 1. A trunk PPP is completely
determined by its number of columns and its intrinsic thickness.
The intrinsic thickness of a general PPP, is the intrinsic thickness
of the trunk PPP obtained after pruning it.
11 / 17
Underlying graph of a trunk PPP
The underlying graph of a trunk PPP is a disjoint union of cycles
of the same even size. It is not injective, two trunk PPP can have
the same underlying graph.
1
2
4
3
3
3
2
1
4
4
1
2
3
4
1
2
1
2
3
4
3
4
4
3
2
1
4
3
2
1
12 / 17
Decomposition of a PPP in trees
1
1
2
3
4
1
2
3
4
2
1
2
3
4
3
4
13 / 17
Decomposition of a PPP in trees
1
1
2
3
4
1
2
3
4
1
2
3
4
3
4
!
tint = 2
2
13 / 17
Decomposition of a PPP in trees
1
1
2
3
4
1
2
3
4
1
2
3
4
3
4
!
tint = 2
2
13 / 17
Decomposition of a PPP in trees
1
2
4
1
2
3
4
1
2
3
!
tint = 2
1
2
3
4
3
4
13 / 17
PPPs : Bijection, enumeration
Theorem
PPPs (except rectangular shaped ones) are in bijection with pairs
consisting of a positive integer corresponding to the intrinsic
thickness and a list of 4-tuples of bicolored ordered trees such that:
I
each 4-tuple is composed of 2 black rooted trees and 2 white
rooted trees,
I
a black vertex is marked in the first 4-tuple.
14 / 17
PPPs : Bijection, enumeration
Theorem
PPPs (except rectangular shaped ones) are in bijection with pairs
consisting of a positive integer corresponding to the intrinsic
thickness and a list of 4-tuples of bicolored ordered trees such that:
I
each 4-tuple is composed of 2 black rooted trees and 2 white
rooted trees,
I
a black vertex is marked in the first 4-tuple.
The sequence enumerating, with respect to the semi-perimeter, the
PPPs of fixed
intrinsic thickness, are counted by the sequence
2n+1
n
(4 − n )n>1 (A008549).
14 / 17
PPPs : Bijection, enumeration
Theorem
PPPs (except rectangular shaped ones) are in bijection with pairs
consisting of a positive integer corresponding to the intrinsic
thickness and a list of 4-tuples of bicolored ordered trees such that:
I
each 4-tuple is composed of 2 black rooted trees and 2 white
rooted trees,
I
a black vertex is marked in the first 4-tuple.
The sequence enumerating, with respect to the semi-perimeter, the
PPPs of fixed
intrinsic thickness, are counted by the sequence
2n+1
n
(4 − n )n>1 (A008549). It counts :
I
“The sum of the areas under all Dyck excursions of length
2n.”
I
“Number of inversions in all 321-avoiding permutations of
[n + 1].”
14 / 17
PPPs : Bijection, generating function.
Theorem
PPPs (except rectangular shaped ones) are in bijection with pairs
consisting of a positive integer corresponding to the intrinsic
thickness and a list of 4-tuples of bicolored ordered trees such that:
I
each 4-tuple is composed of 2 black rooted trees and 2 white
rooted trees,
I
a black vertex is marked in the first 4-tuple.
15 / 17
PPPs : Bijection, generating function.
Theorem
PPPs (except rectangular shaped ones) are in bijection with pairs
consisting of a positive integer corresponding to the intrinsic
thickness and a list of 4-tuples of bicolored ordered trees such that:
I
each 4-tuple is composed of 2 black rooted trees and 2 white
rooted trees,
I
a black vertex is marked in the first 4-tuple.
A• (z• , z◦ ) =
1
1
et A◦ (z• , z◦ ) =
.
1 − z◦ A◦
1 − z• A•
PPP(z• , z◦ ) =
z• ∂z• z• z◦ A• (z• , z◦ )2 A◦ (z• , z◦ )2
.
1 − z• z◦ A• (z• , z◦ )2 A◦ (z• , z◦ )2
15 / 17
PPPs : Bijection, generating function.
Theorem
PPPs (except rectangular shaped ones) are in bijection with pairs
consisting of a positive integer corresponding to the intrinsic
thickness and a list of 4-tuples of bicolored ordered trees such that:
I
each 4-tuple is composed of 2 black rooted trees and 2 white
rooted trees,
I
a black vertex is marked in the first 4-tuple.
A• (z• , z◦ ) =
1
1
et A◦ (z• , z◦ ) =
.
1 − z◦ A◦
1 − z• A•
PPP(z• , z◦ ) =
PPP(z, z) =
z• ∂z• z• z◦ A• (z• , z◦ )2 A◦ (z• , z◦ )2
.
1 − z• z◦ A• (z• , z◦ )2 A◦ (z• , z◦ )2
zC (z)2
z
, where C(z) =
.
1 − 4z
1 − C(z)
15 / 17
Strips : Bijection, generating function
Theorem
Strips (except rectangular ones) are in bijection with pairs
consisting of a positive integer corresponding to the intrinsic
thickness and a cycle of 4-tuples of bicolored ordered trees such
that each 4-tuple is composed of 2 black rooted trees and 2 white
rooted trees.
16 / 17
Strips : Bijection, generating function
Theorem
Strips (except rectangular ones) are in bijection with pairs
consisting of a positive integer corresponding to the intrinsic
thickness and a cycle of 4-tuples of bicolored ordered trees such
that each 4-tuple is composed of 2 black rooted trees and 2 white
rooted trees.
By Pólya’s theory, the generating function of Strips, with respect
to the semi-perimeter, with a fixed intrinsic thickness:
S(z) = −
X ϕ(i)
i>1
i
log(1 − z 2i A(z i )4 ),
where ϕ is the Euler phi function and
A(z) =
1
.
1 − zA(z)
16 / 17
Thank you.
17 / 17