CHAPTER 6
Partitions of a
Convex Polygon
6.1 INTRODUCTION
In our survey we have come across some results on enumeration of noncrossing configurations on the set of vertices of a convex polygon, such as
triangulations and trees. In [19] exact formulae and limit laws are determined for
several parameters of interest by Marc Noy, some results on the enumeration of
chord diagrams (pairings of 2n vertices of a convex polygon by means of n
disjoint pairs) were presented. He also presented limit laws for the number of
components, the size of the largest component and the number of crossings. The
use of generating functions and of a variation of Levy's continuity theorem for
characteristic functions enabled to establish that most of the limit laws presented
here are Gaussian. Michael S. Floater and Tom Lyche [20] provided a new way of
enumerating all “partitions” of a convex polygon of a certain type, i.e., with a
specified number of triangles, quadrilaterals, and so on, which includes Catalan
numbers as a special case. We adopt the techniques of partitioning a polygon from
[19,20] and figure out some results on convex polygons.
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In Section 6.2 we state the definitions of required terminology and develop
results on enumeration of partitions of Convex Polygons which are partitioned
into co-initial polygons and also into non co-initial polygons in Section 6.3.
Finally we present scope for further work and references.
The following vocabulary on partition of a polygon is adopted from the
literature.
6.2 DEFINITIONS AND NOTATION
6.2.1 Polygon: A polygon is a plane figure that is bounded by a closed path or
circuit, composed of a finite sequence of straight line segments.
6.2.2 Convex Polygon: A convex polygon is a polygon whose interior is a convex
set.
6.2.3 Simple Polygon: A simple polygon is a closed polygonal chain of line
segments in the plane which do not have points in common other than the
common vertices of pairs of consecutive segments.
6.2.4 Partition of a polygon: A partition of a polygon P is a set of polygons such
that the interiors of the polygons do not intersect and the union of the polygons is
equal to the interior of the original polygon P.
6.2.5 Segment of a polygon: Let V1 , V2 ,..., Vn be the vertices of a polygon P . If
there exists a line V V such that the line divides P into two polygons P1 , P2 such
i j
that P1 U P2 = P and P1  P2  VV
i j , then the line Vi V j is called segment of the
polygon. This partitioned polygon is denoted by i,j.
6.2.6 r-partitioned polygon: If segments divide the polygon into ‘r’ parts, then
the polygon is called r-partitioned polygon. The number of r-partitioned polygons
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of p is denoted by Pr ( p) . If Vi1V j1 ,Vi2V j2 ,...,Vir1V jr1 segments divide the polygon
into r parts, then this r -partitioned polygon is denoted by i1 , j1; i2 , j2 ;...; ir 1 , jr 1
Note: r - segments divide the polygon into r  1 parts
6.2.7 Co-initial segments: If all the segments start from same vertex of the
polygon, then these lines are called co-initial segments.
6.2.8 Co-initial r-partitioned polygon: If segments divide the polygon into ‘r’
parts and all segments have same initial point, then the polygon is called co-initial
r-partitioned polygon.
6.2.9 Non-co-initial r-partitioned polygon: If segments divide the polygon into
‘r’ parts and no two segments have same initial point, then the polygon is called
non-co-initial r-partitioned polygon.
6.2.10 Co-initial r-sided polygon partition of convex polygon: If a convex
polygon is partitioned into r-sided polygons and all r-sided polygons have same
vertex, then this partitioning is called co-initial r-sided polygon partition of
convex polygon.
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6.3 ENUMERATION OF r – partitioned POLYGONS
Here we make an attempt to study the enumeration of co-initial rpartitioned polygons of a convex polygon. We also prove certain results to
enumerate the number of non co-initial r-partitioned polygons.
Let P be a closed convex polygon with vertices V1 , V2 ,…, Vn . If j  i  2 ,
Vi V j is a segment of P. A collection of segments is said to be co-initial if they
have a common vertex. A segment Vi V j partitions (=divides) P into two parts. We
denote this partition by i, j.
Result 6.3.1: With notation as above, the number of partitions of a n-sided convex
polygon formed by k co-initial segments is n  n 3 Ck  where 2  k  n  3 .
Proof: For definiteness let V1 be the common vertex. Any set K of segments
with common vertex V1 is obtained by the n-3 vertices V2 , V3 ,…, Vn 1 . These can
be chosen in
n 3
Ck ways. Since each of these vertices gives rise to a partition
along with the vertex V1 , the number of partitions is precisely
n 3
Ck .
Similarly for any common vertex Vi , the number of partitions of a n-sided convex
polygon is n 3 Ck .
Hence the total number of partitions of a n-sided convex polygon formed by k coinitial lines is n  n 3 Ck  .
Theorem 6.3.2: The number of partitions of a n-sided convex polygon formed by
one partitioned line is
n
2
n 3
C1
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Proof: Let V1 ,V2 ,...,Vn be the n vertices of a polygon P.
The possible 2-partitioned polygons with one vertex of the partition line at V1 are
1,3
1,4
1,5
Then number is
1,6 …,n-2
n 3
1,n-1
C1
Therefore for any vertex Vi , the number of partitions of a n-sided convex polygon
is n n 3C1 . But every partition line repeats twice like VV
i j and V jVi .
Hence the total number of partitions of a n-sided convex polygon formed by one
partition line is
n
2
n 3
C1
Theorem 6.3.3: The number of co-initial partitions of a n-sided convex polygon is
n  2n2  n  1
Proof: From theorem 6.3.2 we observe that the number of 2-partitioned polygons
of n-sided convex polygon is
n n 3
. C1 .
2
Also from theorem 6.3.1 we observe that co-initial 3-partitioned, 4-partitioned,…,
(n-2)-partitioned polygons of n-sided convex polygon are
n.n 3 C2 , n.n 3 C3 ,…, n.n3 Cn3 respectively.
Therefore the number of co-initial partitions of a n-sided convex polygon is
n n 3
. C1  n.n 3 C2 + n.n 3 C3 +…+ n.n3 Cn3
2
n
 n  n3 C2  n3 C3  ...  n3 Cn3   .n3 C1
2
 n  n3 C0  n3 C1  n3 C2  n3 C3  ...  n3 Cn3 n3 C0 n3 C1  
 n  2n3  1   n  3 

n
 n  3
2
n
 n  3
2
n n2
 2  n  1
2
Example: The number of co-initial partitioned polygons of a hexagon is 24
Let A1 A2 A3 A4 A5 A6 be a hexagon.
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The partitions of a hexagon formed by one segment are
1,3 1,4
1,5
2,4
2,5
2,6
3,5
3,6
4,6
The number of partitions of a hexagon formed by segment line is
6 6 3
C1  9
2
The 3-partitioned polygons of hexagon are
1,3;1,4
1,3;1,5
1,4;1,5
2,4;2,5
2,4;2,6
2,5;2,6
3,5;3,6
3,5;3,1
3,6;3,1
4,6;4,1
4,6;4,2
4,1;4,2
5 ,1;5,
5,1;5,3
5,2;5,3
6,2;6,3
6,2;6,4
6,3;6,4
Therefore the number of co-initial 3-partitioned polygons of a hexagon is
6.63 C2  18
The number of co-initial partitioned polygons of a hexagon

6 6 2
 2  6  1  33
2
Theorem 6.3.4: The number of partitions of a n-sided convex polygon formed by
k non-co-initial lines is
n n 3
C2 k 1
2
Proof: Let V1 ,V2 ,...,Vn be the n vertices of a polygon P.
Let k  1
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From theorem 6.3.2, the number of partitions at V1 is
of partitions is
n
2
n
2
n 3
C1 and the total number
n 3
C1
We assume that the number of partitions of a n-sided convex polygon formed by
m non-co-initial segments is
n
2
n 3
C2 m1
Let k  m  1 and consider the segments Vi1V j1 ,Vi2V j2 ,...,Vim1V jm1 .
Fix Vi ,Vi  2 . Then each of the remaining k segments divides the polygon with
vertices V1 ,V2 ,...,Vn except Vi 1
at V1 in
n 13
C2 m1 ways.
Fix Vi ,Vi 3 . Then each of the remaining k segments divides the polygon with
vertices V1 ,V2 ,...,Vn except Vi 1 ,Vi  2 at V1 in
n  2 3
C2 m1 ways. Proceeding this way
up to the end, we ultimately get that the number of partitions of a n-sided convex
polygon formed by m+1 non-co-initial segments at V1 is
n4
C2m1 n5 C2m1  ... 2k 1 C2 m1
n3 C2 m1
There fore the number of partitions of a n-sided convex polygon formed by m+1
non-co-initial segments is
n n 3
C2 m1
2
Hence the number of partitions of a n-sided convex polygon formed by k non-coinitial segments is
n
2
n 3
C2 k 1
Theorem 6.3.5: The number of non-co-initial partitioned polygons of a n-sided
convex polygon is n.2n5
Proof: From theorem 6.3.4 we observe that non co-initial 2-partitioned, 3-
n 
partitioned,…,   1 -partitioned polygons of n-sided convex polygon are
2 
n n 3
n
n
. C1 , .n 3 C3 , .n 3 C5 … respectively
2
2
2
Hence the number of non-co-initial partitioned polygons of n-sided convex
polygon is
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n n 3
n
n
. C1 + .n 3 C3 + .n 3 C5 +…
2
2
2

n n 3
 C1  n3 C3  n3 C5  ...
2

n n 31
2

2
 n.2n5
Example: The number of non-co-initial partitioned polygons of a 9-sided polygon
is 24
Solution: Let A1 A2 A3 A4 A5 A6 A7 A8 A9 be a 9-sided polygon.
The 2-partitioned polygons with repetition being rounded are
Therefore the number of non-co-initial 2-partitioned polygons is
9 9 3
. C1  27
2
The 3-partitioned polygons are
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Hence the number of non-co-initial 3-partitioned polygons is
9 9 3
. C3  90
2
The 4-partitioned polygons are
103
(repetitions are rounded off)
Hence the number of non-co-initial 4-partitioned polygons is
9 9 3
. C5  27
2
Hence the number of non-co-initial partitioned polygons of 9-sided convex
polygon is
9.295  9.24  144
Theorem 6.3.6: If  r  2  |  n  2  , then the number of co-initial r-sided polygon
partitions of a n-sided convex polygon is
n(n  2)
.
r 2
Proof: Let P be a n-sided convex polygon and r  N such that  r  2  |  n  2  . The
n-sided convex polygon can be partitioned as r-sided polygons.
The left and right r-sided polygons have one segment as a side and remaining
sides are sides of P and remaining r-sided polygons have two segments as sides
and remaining sides are sides of polygon P. The left and right r-sided polygons
have r-1 sides which are sides of polygon P and remaining r-sided polygons have
(r-2)-sides that are sides of polygon P.
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Hence the number of co-initial r-sided polygon partitions of n-sided convex
polygons is
n(n  2)
r 2
Theorem 6.3.7: If r | n the number of r  1 -partitioned polygons of a n-sided
convex polygon partitioned by a closed chain connected with segments is
n n  r 1
.
Cr 1 .
r
Proof: Let V1 , V2 ,..., Vn be n vertices of a polygon P . Let Vi1Vi2 ,Vi2Vi3 ,...,Vir Vi1 be
the closed chain connected with segments.
Let r  3
(1,3 ; 3,5 ; 5,7), (1,3 ; 3,5 ; 5,8), (1,3 ; 3,5 ; 5,9), … , (1,3 ; 3,5 ; 5,2n-1)
(1,3 ; 3,6 ; 6,8), (1,3 ; 3,6 ; 6,9), … , (1,3 ; 3,6 ; 6,2n-1)
…
…
…
…
…
(1,3 ; 3,2n-3 ; 2n-3,2n-1)
(1,4 ; 4,6 ; 6,8), (1,4 ; 4,6 ; 6,9),…, (1,4 ; 4,6 ; 6,2n-1)
…
…
…
…
…
(1,4 ; 4,2n-3 ; 2n-3,2n-1)
…
…
…
…
…
(1,2n-5 ; 2n-5,2n-3 ; 2n-3,2n-1)
The number of above partitioned polygons are
n4
C2 . But every polygon repeats
thrice
Hence the number of above partitioned polygons are
n n4
. C2 .
3
When r  k we can similarly prove that the number of partitioned polygons are
n  r 1
Cr 1 . But every polygon is repeated r  times.
Hence the number of partitioned polygons are
n n  r 1
.
Cr 1 .
r
Hence the number of r  1  partitioned polygons of n-sided convex polygon
partitioned by the closed chain connected with segments is
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n n  r 1
.
Cr 1 .
r
SCOPE FOR FURTHER WORK
Motivated by the above out come of our investigation on enumeration of
integer partitions and partitions of convex polygons, we find interesting to
continue our pursuit along the following lines.
 Extension of these results to overpartition pairs.
 Relation between r-partitions of different positive integers for which
number of r-partitions is maximum.
 Developing programs for the enumeration of r  partitions of n from the
new results in MATLAB.
 Preparation of programs for LS (List Schedulings) algorithm by employing
built in function 'fmincon' in MATLAB for r – partition optimizing some
objective functions like maximizing the sum of the smallest k parts,
minimizing the ratio of the largest to the smallest part in which 1  k  r .
 Computational aspects of r  partitions of convex polygons with the aid of
NEURAL NETWORKS aiming at traffic control problems.
 L2 metric performance on the LS (List Schedulings) algorithm.
 Enumeration of co-initial and non co-initial r-partitioned polygons,
partitioned polygons in a second layered polygon [9].
 Search for any possible link between integer partitions and partitions of
convex polygons.
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