Difference between revisions of "Partition"
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+ | {{MSC|54B}} | ||
+ | ==Of a topological space== | ||
A closed set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p0717401.png" /> in a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p0717402.png" /> that partitions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p0717403.png" /> between two given sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p0717404.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p0717405.png" /> (or, in other words, separates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p0717406.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p0717407.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p0717408.png" />), i.e. such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p0717409.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174011.png" /> are disjoint and open in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174014.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174016.png" /> are open in all of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174017.png" />). A partition is called fine if its interior is empty. Any binary [[Decomposition|decomposition]] (i.e. a partition consisting of two elements) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174018.png" /> of a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174019.png" /> defines a fine partition in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174020.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174021.png" /> is the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174022.png" />, which is the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174023.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174024.png" />, in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174025.png" /> is the open kernel (cf. [[Kernel of a set|Kernel of a set]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174027.png" />. The converse is also true. In essence, the concept of a partition between sets leads to the concept of connectedness. The converse also applies: A space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174028.png" /> is disconnected if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174029.png" /> is a partition between non-empty sets. | A closed set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p0717401.png" /> in a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p0717402.png" /> that partitions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p0717403.png" /> between two given sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p0717404.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p0717405.png" /> (or, in other words, separates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p0717406.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p0717407.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p0717408.png" />), i.e. such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p0717409.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174011.png" /> are disjoint and open in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174014.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174016.png" /> are open in all of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174017.png" />). A partition is called fine if its interior is empty. Any binary [[Decomposition|decomposition]] (i.e. a partition consisting of two elements) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174018.png" /> of a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174019.png" /> defines a fine partition in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174020.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174021.png" /> is the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174022.png" />, which is the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174023.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174024.png" />, in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174025.png" /> is the open kernel (cf. [[Kernel of a set|Kernel of a set]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174027.png" />. The converse is also true. In essence, the concept of a partition between sets leads to the concept of connectedness. The converse also applies: A space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174028.png" /> is disconnected if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174029.png" /> is a partition between non-empty sets. | ||
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One readily sees that every partition is a separator and that every separator is a cut, and the following examples show that the notions are in general distinct: the open interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174048.png" /> is a separator between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174049.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174050.png" /> in the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174051.png" />, but not a partition; in the well-known subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174052.png" /> of the Euclidean space, the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174053.png" /> is a cut but not a separator between the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174054.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174055.png" />. | One readily sees that every partition is a separator and that every separator is a cut, and the following examples show that the notions are in general distinct: the open interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174048.png" /> is a separator between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174049.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174050.png" /> in the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174051.png" />, but not a partition; in the well-known subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174052.png" /> of the Euclidean space, the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174053.png" /> is a cut but not a separator between the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174054.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174055.png" />. | ||
+ | {{MSC|11P}} | ||
+ | ==Of a positive integer== | ||
A partition of a positive integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174056.png" /> is a decomposition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174057.png" /> as a sum of positive integers. For example, the partitions of 4 read: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174058.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174059.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174060.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174061.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174062.png" />. The number of different partitions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174063.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174064.png" />. So, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174065.png" />. L. Euler gave a non-trivial recurrence relation for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174066.png" /> (see [[#References|[a1]]]) and Ramanujan discovered the surprising congruences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174067.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174068.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174069.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174070.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174071.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174072.png" />), and others. He also found the asymptotic relation | A partition of a positive integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174056.png" /> is a decomposition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174057.png" /> as a sum of positive integers. For example, the partitions of 4 read: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174058.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174059.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174060.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174061.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174062.png" />. The number of different partitions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174063.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174064.png" />. So, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174065.png" />. L. Euler gave a non-trivial recurrence relation for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174066.png" /> (see [[#References|[a1]]]) and Ramanujan discovered the surprising congruences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174067.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174068.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174069.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174070.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174071.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174072.png" />), and others. He also found the asymptotic relation | ||
Revision as of 18:26, 21 November 2014
2020 Mathematics Subject Classification: Primary: 54B [MSN][ZBL]
Of a topological space
A closed set in a topological space
that partitions
between two given sets
and
(or, in other words, separates
and
in
), i.e. such that
, where
and
are disjoint and open in
,
,
(
and
are open in all of
). A partition is called fine if its interior is empty. Any binary decomposition (i.e. a partition consisting of two elements)
of a space
defines a fine partition in
:
is the boundary of
, which is the boundary of
, where
, in which
is the open kernel (cf. Kernel of a set) of
,
. The converse is also true. In essence, the concept of a partition between sets leads to the concept of connectedness. The converse also applies: A space
is disconnected if
is a partition between non-empty sets.
Comments
Related notions in this context are those of a separator and of a cut.
If and
are disjoint subsets of a space
, then a separator between
and
is a set
such that
with
and
disjoint and open in
, and
and
. So a partition is a closed separator.
A set is a cut between
and
if
intersects every continuum that intersects both
and
.
One readily sees that every partition is a separator and that every separator is a cut, and the following examples show that the notions are in general distinct: the open interval is a separator between
and
in the interval
, but not a partition; in the well-known subspace
of the Euclidean space, the point
is a cut but not a separator between the points
and
.
2020 Mathematics Subject Classification: Primary: 11P [MSN][ZBL]
Of a positive integer
A partition of a positive integer is a decomposition of
as a sum of positive integers. For example, the partitions of 4 read:
,
,
,
,
. The number of different partitions of
is denoted by
. So,
. L. Euler gave a non-trivial recurrence relation for
(see [a1]) and Ramanujan discovered the surprising congruences
(
),
(
),
(
), and others. He also found the asymptotic relation
![]() |
where . Later this was completed to an exact series expansion by H. Rademacher (see [a2]).
One can also distinguish other partitions, having particular properties, such as the numbers in the decomposition being distinct (see [a3]). See also Additive number theory; Additive problems.
References
[a1] | G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Oxford Univ. Press (1979) pp. Chapt. XVI |
[a2] | T.M. Apostol, "Modular functions and Dirichlet series in number theory" , Springer (1976) |
[a3] | G.E. Andrews, "The theory of partitions" , Addison-Wesley (1976) |
[a4] | R. Engelking, "Dimension theory" , North-Holland & PWN (1978) pp. 19; 50 |
Partition. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Partition&oldid=34709