![IOSR Journal of Mathematics (IOSR-JM)
e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 13, Issue 1 Ver. IV (Jan. - Feb. 2017), PP 44-49
www.iosrjournals.org
DOI: 10.9790/5728-1301044449 www.iosrjournals.org 44 | Page
Enumeration Class of Polyominoes Defined by Two Column
Eman f. Mohommed1
, Haslinda Ibrahim2
, Nazihah Ahmad3
1
School of Quantitative Sciences, College of Arts and Sciences, Universiti Utara Malaysia 06010 Sintok,
Kedah, Malaysia.
2
School of Quantitative Sciences, College of Arts and Sciences, Universiti Utara Malaysia 06010 Sintok,
Kedah, Malaysia.
3
School of Quantitative Sciences, College of Arts and Sciences, Universiti Utara Malaysia 06010 Sintok,
Kedah, Malaysia.
Abstract: Abacus diagram is a graphical representation for any partition µ of a positive integer t. This study
presents the bead positions as a unite square in the graph and de ne a special type of e-abacus called nested
chain abacus 𝔑 which is represented by the connected partition. Furthermore, we redefined the polyominoes as
a special type of e-abacus diagram. Also, this study reveals new method of enumerating polyominoes special
design when e=2.
Mathematics Subject Classification: 05A15
Keywords: Abacus, Polyominoes, n-connected square, enumeration, Free enumeration, Combinatorial.
I. Introduction
The enumeration of polyominoes is one of the studied subject in combinatorics. In 1953,
Soloman Golomb introduced the term polyomino, and it was popularized by Gandner in
1957. Polyominoes is defined as a finite collection of n-monomino in which the monomino is a
unite square in the plan Z × Z. The general enumeration problem of polyominoes has not been
solved yet, and it remains open to solution. There are three common ways of distinguishing
polyominoes from n-monomino for enumeration which are, free polyominoes, one side and
fixed polyominoes. Free polyominoes or P (n) are considered different distinct when none is a
rotation or translation of another while their location or orientation is not important in the
plan Fixed polyominoes are consider distinct when if they have different shape [9], [10], [14]. To
the best of our knowledge, no one has attempted to find the enumeration of fixed polyominoes or
n-connected square with 2 columns. Hence, this study proposes new method to find
polyominoes by using nested chain abacus, and enumeration of n-connected square
(polyominoes) which is a representation of connected bead positions sequences.
II. Preliminary Notes
In this section, we will redefine polyominoes as nested chain abacus. The first step is to
examine the structure of abacus diagram, and, in the subsequent section, we discuss
some important definitions. Nested chain abacus is a graphical representation of non-
increasing sequence µ(e,r)
= ( µ1 , µ2, ..., µb ) called connected partition. To define the
nested chain abacus 𝔑, we must first determine the value of e. where e represents the
number of the columns in the abacus which are labelled from 0 to e-1 and e ≥ 1. The
bead positions are located across the columns on the nested chain abacus which are
labelled from left to right and continued to lower from top starting with 0, the bead
positions ne, ne + 1, ne + 2, (n+1)e -1 are located in row n in the nested chain abacus.
The process to define abacus is much easier if it involves usage of a graph. We can
think of our abacus in the fourth quadrant of a standard two-dimension plane and
any bead as a unit square in two-dimension plane. In following section, we define
some important definitions foundation of this work.](https://image.slidesharecdn.com/i1301044449-170708045003/85/Enumeration-Class-of-Polyominoes-Defined-by-Two-Column-1-320.jpg)
Enumeration Class of Polyominoes Defined by Two Column
- 1. IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 13, Issue 1 Ver. IV (Jan. - Feb. 2017), PP 44-49 www.iosrjournals.org DOI: 10.9790/5728-1301044449 www.iosrjournals.org 44 | Page Enumeration Class of Polyominoes Defined by Two Column Eman f. Mohommed1 , Haslinda Ibrahim2 , Nazihah Ahmad3 1 School of Quantitative Sciences, College of Arts and Sciences, Universiti Utara Malaysia 06010 Sintok, Kedah, Malaysia. 2 School of Quantitative Sciences, College of Arts and Sciences, Universiti Utara Malaysia 06010 Sintok, Kedah, Malaysia. 3 School of Quantitative Sciences, College of Arts and Sciences, Universiti Utara Malaysia 06010 Sintok, Kedah, Malaysia. Abstract: Abacus diagram is a graphical representation for any partition µ of a positive integer t. This study presents the bead positions as a unite square in the graph and de ne a special type of e-abacus called nested chain abacus 𝔑 which is represented by the connected partition. Furthermore, we redefined the polyominoes as a special type of e-abacus diagram. Also, this study reveals new method of enumerating polyominoes special design when e=2. Mathematics Subject Classification: 05A15 Keywords: Abacus, Polyominoes, n-connected square, enumeration, Free enumeration, Combinatorial. I. Introduction The enumeration of polyominoes is one of the studied subject in combinatorics. In 1953, Soloman Golomb introduced the term polyomino, and it was popularized by Gandner in 1957. Polyominoes is defined as a finite collection of n-monomino in which the monomino is a unite square in the plan Z × Z. The general enumeration problem of polyominoes has not been solved yet, and it remains open to solution. There are three common ways of distinguishing polyominoes from n-monomino for enumeration which are, free polyominoes, one side and fixed polyominoes. Free polyominoes or P (n) are considered different distinct when none is a rotation or translation of another while their location or orientation is not important in the plan Fixed polyominoes are consider distinct when if they have different shape [9], [10], [14]. To the best of our knowledge, no one has attempted to find the enumeration of fixed polyominoes or n-connected square with 2 columns. Hence, this study proposes new method to find polyominoes by using nested chain abacus, and enumeration of n-connected square (polyominoes) which is a representation of connected bead positions sequences. II. Preliminary Notes In this section, we will redefine polyominoes as nested chain abacus. The first step is to examine the structure of abacus diagram, and, in the subsequent section, we discuss some important definitions. Nested chain abacus is a graphical representation of non- increasing sequence µ(e,r) = ( µ1 , µ2, ..., µb ) called connected partition. To define the nested chain abacus 𝔑, we must first determine the value of e. where e represents the number of the columns in the abacus which are labelled from 0 to e-1 and e ≥ 1. The bead positions are located across the columns on the nested chain abacus which are labelled from left to right and continued to lower from top starting with 0, the bead positions ne, ne + 1, ne + 2, (n+1)e -1 are located in row n in the nested chain abacus. The process to define abacus is much easier if it involves usage of a graph. We can think of our abacus in the fourth quadrant of a standard two-dimension plane and any bead as a unit square in two-dimension plane. In following section, we define some important definitions foundation of this work.
- 2. Enumeration Class of Polyominoes Defined by Two Column DOI: 10.9790/5728-1301044449 www.iosrjournals.org 45 | Page Definition 2.1. The bead positions βi is a sequence of four vertices with length 1, These vertices are connected by horizontal or vertical vertex edges with length 1 such that any two vertices are collinear. The vertices are a coor- dinates in the fourth quadrant of two-dimension plane, thus βi = ne + c = {( c, −n ), ( c + 1, −n ), ( c, −n − 1 ), ( c + 1, −n − 1 ) |0 ≤ c ≤ e − 1 and 0 ≤ n ≤ r-1}
- 3. Enumeration Class of Polyominoes Defined by Two Column DOI: 10.9790/5728-1301044449 www.iosrjournals.org 46 | Page III. Some combinatorial result
- 4. Enumeration Class of Polyominoes Defined by Two Column DOI: 10.9790/5728-1301044449 www.iosrjournals.org 47 | Page
- 5. Enumeration Class of Polyominoes Defined by Two Column DOI: 10.9790/5728-1301044449 www.iosrjournals.org 48 | Page IV. Conclusion Reference
- 6. Enumeration Class of Polyominoes Defined by Two Column DOI: 10.9790/5728-1301044449 www.iosrjournals.org 49 | Page