# T

## Tetrahedral Numbers

Integers that can be expressed as a tetrahedral array in 3-space.
Another charcterization is as the
binomial coefficient * C(n,4) * (for * n > 3 *).

An explicit formula for *T*_{n} in terms of *n* is:

*T*_{n} = n(n+1)(n+2)/6

## Thrackle

A graph formed by a collection of curves in the plane,
with the property that the curves corresponding to
any pair of edges meet in a single point,
which is either an endpoint of both edges or
a place where the two curves cross.
Every planar straight line graph thrackle has at most
as many edges as vertices; Conway's thrackle conjecture
states that this is true even for curved thrackles.

## Transitive

In general, transitivity is a property of a relation: if whenever
* a~b * and *b~c *, then * a~c *, the relation
is transitive. Equality is a transitive relation, so is (set-wise)
containment.

In group theory, when a group has a
permutation action on a set, the action is called transitive if there
are group elements whose permutation action is to exchange any given
pair of elements of the set. Such an action is called doubly-transitive
if any two ordered pairs can be exchanged by the permutation action of
some group element, triply-transitive if any two ordered triples can be
exchanged, etc.

## Tree

A connected graph having no cycles.

## Triangular Numbers

Integers that can be expressed as triangular arrays of dots.
Another charcterization is as the
binomial coefficient * C(n,4) * (for * n > 3 *).

An explicit formula for the *n*th triangular number is:

*T*_{n} = n(n+1)/2

## Triangulation

A simplicial complex covering all of a given region in R^d.

## Turan's Theorem

Let *t* and *n* be positive integers with *t* at
least equal to 2, and *n* greater than or equal to *t*.
The maximum number of edges of a graph of order *n* that doesn't
contain a complete subgraph of order t is

where the *n*_{i} form a partition of *n* into *t-1*
parts which are as equal as possible.

Furthermore, the complete (t-1)-partite graph with parts of size
*n*_{1},n_{2},...n_{t-1}
is the only graph whose number of edges is
equal to the bound above but still doesn't contain a complete subgraph
of order *t*.

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An On-line Dictionary of Combinatorics