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 *i*_{1},
*i*_{2}, ..., *i*_{s} of positive integers with sum *r*.

* __Enumeration__: The
number of partitions of *r* is *p*(*r*)
~ exp[π(2*r*/3)^{1/2}]/(4·3^{1/2}*r*)
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 {*O*_{i}}
of *M*, a partition of unity subordinate to it is a collection {*f*_{i}}
of smooth real functions, such that (i) Supp(*f*_{i}) ⊂ *O*_{i},
(ii) 0 ≤ *f*_{i} ≤ 1,
(iii) ∑_{i}* f*_{i}
= 1.

* __Condition__: It exists
for any {*O*_{i}} if the closure
of each *O*_{i} 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* → (*l*)^{n}_{m}, with *n* positive
integer and *k*, *l*, *m* cardinals.

* __Simplest example__: 6 → (3)^{2}_{2};
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