Partitions| Partitions |
Of an Integer > s.a. lattice [of partitions of an integer];
Multinomial Coefficients.
$ Def: A partition of a positive
integer r is an unordered sequence i1,
i2, ..., is
of positive integers with sum r.
* Enumeration: The number of partitions
of r is p(r)
~ exp[π(2r/3)1/2]/(4·31/2r)
for r → ∞; For small r, the exact values are
r:
1 2 3 4 5 6 7 8 9 10
p(r): 1 2 3 4 7 11 15 22 30 42.
@ General references: in Ostmann 56;
Andrews 76;
Okounkov mp/03 [random];
Lee DM(06).
@ Counting problem: Rovenchak a1603-conf [statistical mechanical approach].
> Online resources:
see MathWorld page;
Wikipedia page.
Of a Set
* Idea: A partition of a set
X is a collection of disjoint subsets whose union is X.
@ References: Cameron DM(05) [associated with permutations].
Partition of Unity > s.a. mixed states.
$ Def: Given a manifold M and
a locally finite open cover {Oi}
of M, a partition of unity subordinate to it is a collection
{fi} of smooth real functions,
such that (i) Supp(fi)
⊂ Oi,
(ii) 0 ≤ fi
≤ 1, (iii) ∑i
fi = 1.
* Condition: It exists for
any {Oi} if the closure
of each Oi is compact
[@ Kobayashi & Nomizu 69].
* Relationships: Existence of
a partition of unity subordinate to any open cover is equivalent to paracompactness.
Partition Relation
* Idea: A central notion in combinatorial set theory.
$ Def: If A is a set of cardinality
k and the set [A]n of unordered
n-element subsets of A is partitioned into m pieces, then there
is a set B ⊂ A with cardinality l such that all elements of
[B]n lie in the same piece of the partition
[B is called homogeneous for the partition] (Ramsey theorem).
* Notation: Expressed as k with n
→ (l)nm,
a positive integer and k, l, m cardinals.
* Simplest example:
6 → (3)22;
If all edges of a complete 6-vertex graph are 2-colored, there is at least one
monochromatic triangle.
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send feedback and suggestions to bombelli at olemiss.edu – modified 27 nov 2017