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* with *n*
→ (*l*)^{n}_{m},
a 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