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].

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 BA 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.