International Journal of Advanced Computing, ISSN: 2051-0845, Vol.46, Issue.3
1278
Partition of a Set with N Elements into K Blocks
with Number of Elements in Accordance with the
Composition of Number N As a Sum of Any K
Natural Summands (Another Representation of
Stirling Number)
Pellumb Kllogjeri
Adrian Kllogjeri
Statistics and Graph Theory University “Aleksander
Xhuvani”, Elbasan, Albania Faculty of Natural
Sciences, Department of Mathematics and Informatics
Email: [email protected]
MSc in Statistics: University of Kent, UK
Actuarial Programmer, Company: AIG Europe Ltd,
London
Email: [email protected]
ABSTRACT
In the book about the generating functions (Generatingfunctionology) that is a bridge between discrete mathematics and
continuous analysis, Prof. Herbert S. Wilf of University of Pennsylvania presents many of their applications. Among
many other topics, treated in this book, is the one concerning the partition of a set S into blocks or parts and the
respective number of partitions. The concept on the partition of a set S into k blocks is very general, meaning that each
of k blocks is of any cardinality (the numbers of the elements belonging to the blocks can be different or equal).
In this paper we consider the case when the number of partitions of the set [n] into k blocks and the numbers of the
elements in each box are different from one another, also the combined case: the numbers of the elements in each block
can be different or equal.
Lastly we have presented a new formula for computing the number of partitions of the set [n] into k blocks with respect
to the representation of the number n as a sum of k natural summands (different or equal). The Stirling numbers of the
second kind, or Stirling partition numbers, describe the number of ways a set with n elements can be partitioned into k
disjoint, non-empty subsets. Common notations are S (n, k) and,
-. The study is finalized with an extraordinary identity
between the second type Stirling number and the number of partitions of number n in accordance with the composition of
number n with summands. We present a second way for computing the number of partitions of number n leading to the
same result achieved by Stirling. The paper is a specific contribution for specific needs. It is an answer to different and
specific requirements of specialists and people of different fields. Here will find answer all of those who are concerned
about the number of partitions of number n into k parts with the same number of elements, with different number of
elements or, several parts having the same number of elements and the others different numbers of elements.
Keywords-partition of a set, partition of number n in accordance with its composition with summands, classes, subpartition of rank r, index of sub-partition
1. PRELIMINARIES
We know from the theory of generating functions that the number of partitions of the set [n] into any k classes , denoted
, - and called the Stirling number of second kind([7] and [6]), is
{ }
∑
(
)
(
(1)
)
1.1 Partition Into K Blocks Of Rank R
If n=k∙r, where n,k,r ∈ N then, the number of partitions of the set with n elements into k blocks with r elements each one
of them (blocks of rank r) is denoted and is:
[
(
)
(
(
)
)
(
]
(
)
)
(
)
(
)
( )
Using proper formula for the binomial coefficients the expression (2) takes the form:
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(2)
International Journal of Advanced Computing, ISSN: 2051-0845, Vol.46, Issue.3
[
]
(
1279
)
(3)
( )
1.2 Partition into k Blocks of different sizes
The expression for the number of partitions of the set with n elements into k blocks with
different), respectively, can be transformed as following:
*
( ) (
) (
)
(
(
)(
elements (all are
+
) ( )
(
) (
) (
)(
) (
)
)
(4)
The last expression for the number of partitions of the set [n] into k classes with
elements, respectively, is the
same number of different permutations of n objects, where there are
indistinguishable objects of type 1,
indistinguishable objects of type 2, …….., and indistinguishable objects of type k .
1.3 Combined Case
Suppose that
where, ∈
and the summands are different or equal. Let assume that the
composition of number n contains different and equal summands as well. This means that there are summands satisfying
the condition:
*
+.
Further, let assume that there are different groups of summands such that the summands within the group are equal to one
another. The natural numbers (summands) equal to one another will be grouped, i.e., we form groups of equal summands.
The groups of equal summands in the composition of number n, will be put in the beginning of the sum (having no
problem with the order of the groups with equal summands). Let these groups be ordered like this:
(5)
The groups of equal summands in the composition of number n can also be put after the summands that are all different.
The result concerning the number of partitions is the same, as will be seen in another assertion.
So, consider the case when the sequence of the groups with equal summands will be followed by the non-equal
summands (having no problem regarding their order, because we know that the addition operator defined in the set of
natural numbers satisfies the associative and commutative property). On the other side, the way the summands in the
composition of the number n are added does reflect the way that the elements in quantities are distributed into k boxes.
It is clear that, it is not important which quantity of elements in which box is put into. In this case we have partitions into
blocks where the blocks containing equal numbers of elements are put in the beginning whereas, the blocks with different
numbers of elements follow them.
1.3.1 Definition 1
A group of classes (blocks) of a partition P is called sub-partition of P. In our case we have different sub-partitions. To
make distinction between them we give:
1.3.2 Definition 2
A group of blocks of a partition P containing the same number r of elements (r < n) is called sub-partition of rank r of the
partition P.
2.
PARTITION IN ACCORDANCE WITH THE COMPOSITION OF NUMBER
The meaning of the sequence of the equalities (5) is as following. In a fixed partition, built in accordance with the given
composition of the number n, there is a sub-partition of
blocks with equal numbers of elements, namely
(subpartition of rank ), there is a sub-partition of
blocks with
elements into each one of them (sub-partition of rank
), ……….., there is a sub-partition of
blocks with
elements into each one of them (sub-partition of rank ) .
All the other blocks have different numbers of elements.
Combining the results of (3) and (4), the number of different partitions of n distinct elements into k blocks in accordance
with the given composition of n is computed as in the following corollary:
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International Journal of Advanced Computing, ISSN: 2051-0845, Vol.46, Issue.3
1280
2.1 Corollary
Let be given that
where rj ∈ N and some of its summands form groups of equal numbers
(suppose we have l groups of equal numbers):
Then the
number of different partitions of the number n into k blocks with
elements, respectively is:
*
+
∑
∏
(
)
(6)
where,
{
(7)
The natural numbers , which are indices of the summands
in the composition of number n, are defined by the
following expression in such a way that they satisfy the respective formula (2) for the number of partitions of a set into
blocks with the same number of elements for each sub-partition of rank :
{
(8)
The indices
that are natural numbers are called indices of sub-partitions having
blocks,
respectively with the same number of elements within the sub-partition. The indices defined by formula (8) reflect the
case when the number n is composed in the form:
, meaning that there
are
summands equal to the natural numbers
, respectively and the rest of them,
,
are all different.
Having into consideration this composition of the number n and the expressions (6), (7) and (8), the number of partitions
of n into k classes (blocks) can be transformed as following:
*
+
∑
∏
(
(
( )
(
)
(
)
(
)
)
(
)
(
)
∑
(
)
∑
)
(
∑
(
)
)
∑
(
)
∑
( )
(
)
(
) (
)
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International Journal of Advanced Computing, ISSN: 2051-0845, Vol.46, Issue.3
(
( ) (
)
)
(
)
∑
…
(
(
)(
) (
∑
(
∑
) (
)
∑
1281
)
)
(
)
(
)
∑
(
)(
)
( )
*
+
(
)
(
)
(
(9)
)
Note: The so far conclusions got on the number of partitions of the set with n elements into k blocks with the same
number of elements, and when they are all different from one another, establish the base for computing the number of
partitions of a set with n elements into k blocks in accordance with the composition of the number n as a sum of any k
natural summands.
From corollary 2.1, if
where,
∈
and some of its summands form groups of equal numbers:
,
then the number of different partitions of number n into k blocks with
elements, respectively is :
∑
*
where,
∈
+
∏
(
)
(11)
and
,
As aforementioned, the natural numbers , that are indices of the summands in the composition of the number n , are
defined in such a way that they satisfy the respective formula (3) for the number of partitions of a set into classes with the
same number of elements for each sub-partition:
{
Now, let us consider the compositions of the natural number n as a sum of k natural numbers and, let assume that all
possible and different compositions of the natural number n as a sum of k natural summands (in the sum can be found
different and equal numbers) are those presented in the system of the identities (s different ways):
{
∈
(12)
All the compositions are different and each one of them represents a way of how n elements are distributed in k classes or
blocks. In other words, in the system (12) we have all the ways that the set of n elements can be partitioned into k classes.
All the ways are different from one another. So, in the system (12) are shown all the ways (s ways in all) that number n is
composed as sum of k numbers. The respective partitions of number n related to the compositions of number n we call:
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International Journal of Advanced Computing, ISSN: 2051-0845, Vol.46, Issue.3
1282
partitions in accordance with the composition of number n as a sum of k natural summands. The compositions of number
n can contain equal numbers:
for ∈ *
+
∈*
+.
The natural numbers (summands) equal to one another will be grouped in groups, i.e., we form groups with equal
summands. In such cases, the groups of equal summands in the composition of number n, will be put at the beginning of
the sum (having no problem regarding the order of the groups with equal summands). The non-equal summands will be
added after the groups of equal summands, having, also, no problem regarding their order, because we know that the
addition operator defined in the set of natural numbers satisfies the associative and commutative property.
On the other side, the way the summands in the composition of the number n are added reflects the way the elements of
quantities
are distributed into k boxes, and it is clear that it is not important what quantity of elements is put into the
first box and what quantity of elements is put into the second box and so on.
Taking into consideration the result achieved in (10) and s different compositions of the number n as sum of k summands
in (12) we get s different types of partitioning of number n. Each type corresponds to one of the compositions in (12). In
the case of the composition of number n in the form:
we have the partitions of n elements
into k classes where, the first class has
elements, the second class has
elements, ……, the k-th class has
elements. Some of
(1 ≤ j ≤ k) can be equal to one another. In this case the number of different partitions of n elements
into k classes is in accordance with the first composition of n and it is:
∑
∏
(
)
(13)
Similarly, in the case of the composition of number n in the form
we have partitions into k
classes also, but the first class has
elements, the second class has
elements, ….., the k-th class has
elements, where some of them can be equal to each other. In this case the number of different partitions of n elements
into k classes is in accordance with the second composition of n and the number of partitions is:
∑
∏
(
)
(14)
The same thing holds true for the other types of compositions of number n up to the composition of type:
for which the number of partitions is:
∑
∏
(
)
(15)
Definitely, the number of all partitions of number n into k classes, where partitions are in accordance with the
composition of number n as a sum of any k natural summands, is:
∑
[
]
(
∑
*∏
(
)+
(16)
)
2.2 Theorem
If all the possible and different compositions of the natural number n as a sum of k natural summands are those shown in
the system of the equalities (12) then the number of all partitions of number n into k classes, where partitions are in
accordance with the composition of number n as a sum of any k natural summands, is equal to the Stirling number of the
second kind.
That is, holds true the identity:
∑
∑
*∏
(
)+
∑
(
)
(
)
where,
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(17)
International Journal of Advanced Computing, ISSN: 2051-0845, Vol.46, Issue.3
1283
,
We have used comma between indices in order to make the distinction between the natural numbers
Here, the expression (10) is adopted in accordance with the compositions of
number n as in (11). Recall that the natural numbers , that are indices of the summands
in the composition of the
number n, are such that:
{
Based on the order of the equalities in the expression for
, it is clear that:
Proof: The theorem will be proved by comparing the sets that are represented by the quantities present in the above
identity. The right side of the identity (17) represents the number of all different partitions of the set [n] into k classes
including all the types of k classes. The left side of (17) represents the general number of the partitions of the same set [n]
, but based on the classification of the classes in accordance with the different ways of the composition of the natural
number n as sum of k summands.
Having into consideration the system of the identities (12) where we have all possible compositions of number n which
are all different from one another, then the left side of (17) is the sum of the numbers of partitions into k classes and
grouped in accordance with s different ways of the compositions of the natural number n. We repeat that the right side of
(17) is the general number of the partitions of n into k classes, but not grouped in accordance with the above
compositions of number n. The identity (17) reflects two ways of computing the number of partitions of the set with n
objects into k classes: one way, by counting the partitions in accordance with the different ways of the composition of the
natural number n as sum of k summands and, the other way, by the method of generating functions. Definitely, the above
identity is proved.//
Note: The method that is applied to prove this theorem is unusual, it is not similar to the methods used in proving a
mathematical statement. For this reason this theorem is a special one and of special interest.
3. CONCLUSIONS AND ADVANTAGES
The case when the cardinal of a set S (number n) is expanded as a sum of k natural summands is of specific importance.
We have partitioned the set of n elements into k classes in accordance with the particular composition of number n. It is
considered the combined case: partition of of the set [n] into k classes when number n is sum of k natural summands
some of which are equal to one another. We proved the special theorem about an unusual identity, the proof of which is
based on comparing the sets that are represented by the quantities present in the identity (17). In one side of the identity
is Stirling number and on the other side is the new representation. By expressing the Stirling number in accordance with
each composition of number n as a sum of k natural numbers, represented by the left side of the identity (17), we have at
hand all the numbers of s different k-partitions.
The paper is an answer to all of those who are concerned about the number of partitions of number n into k parts with the
same number of elements, with different number of elements or several parts having the same number of elements and
the others different numbers of elements. They have to look at different parts of the new formula (LHS of (17)) and use
the one they need.
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[1]
Anders Bjorner, Richard P. Stanley, A Combinatorial Miscellany, electronic version, 2010 (pp.6-23).
[2]
Kenneth P. Bogart, Combinatorics Through Guided Discovery, 2004, (pp. 57-80)
[3]
Leo Moser, An Introduction to the Theory of Numbers, ISBN 978-1-931705-01-1, published by The Trillia
Group, 2004,(pp.1-6)
[4]
Regnaud B. J. T. Allenby, A. B. Slomson, How to Count-An Introduction to Combinatorics, Second Edition,
2011 by Taylor and Francis Group LLC, Printed in USA, International Standard Book Number 13:978-1-42008261-6(pp.51-68)
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International Journal of Advanced Computing, ISSN: 2051-0845, Vol.46, Issue.3
1284
[5]
Rosen Kenneth H., Discrete Mathematics and its Applications, Fifth Edition, Published by McGraw-Hill, ,ISBN
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[6]
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edition (Chapters on Partitions, Stirling's Approximation Partitions and Generating Functions).
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Wilf Herbert S., Generatingfunctionology, Second Edition, Copyright 1990,1994 by Academic Press, Inc (pp.223)
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