Interval order
In mathematics, especially order theory, the interval order for a collection of intervals on the real line is the partial order corresponding to their left-to-right precedence relation—one interval, I1, being considered less than another, I2, if I1 is completely to the left of I2. More formally, a poset is an interval order if and only if there exists a bijection from
to a set of real intervals, so
, such that for any
we have
in
exactly when
. Such posets may be equivalently characterized as those with no induced subposet isomorphic to the pair of two element chains, the
free posets .[1]
The subclass of interval orders obtained by restricting the intervals to those of unit length, so they all have the form , is precisely the semiorders.
The complement of the comparability graph of an interval order (, ≤) is the interval graph
.
Interval orders should not be confused with the interval-containment orders, which are the containment orders on intervals on the real line (equivalently, the orders of dimension ≤ 2).
Interval dimension
The interval dimension of a partial order can be defined as the minimal number of interval order extensions realizing this order, in a similar way to the definition of the order dimension which uses linear extensions. The interval dimension of an order is always less than its order dimension,[2] but interval orders with high dimensions are known to exist. While the problem of determining the order dimension of general partial orders is known to be NP-complete, the complexity of determining the order dimension of an interval order is unknown.[3]
Combinatorics
In addition to being isomorphic to free posets, unlabeled interval orders on
are also in bijection with a subset of fixed point free involutions on ordered sets with cardinality
.[4] These are the involutions with no left or right neighbor nestings where, for
an involution on
, a left nesting is an
such that
and a right nesting is an
such that
.
Such involutions, according to semi-length, have ordinary generating function [5]
.
Hence the number of unlabeled interval orders of size is given by the coefficient of
in the expansion of
.
1, 2, 5, 15, 53, 217, 1014, 5335, 31240, 201608, 1422074, 10886503, 89903100, 796713190, 7541889195, 75955177642, … (sequence A022493 in OEIS)
Notes
References
- Lua error in package.lua at line 80: module 'strict' not found.
- Lua error in package.lua at line 80: module 'strict' not found.
- Lua error in package.lua at line 80: module 'strict' not found.
- Lua error in package.lua at line 80: module 'strict' not found..
- Gunther Schmidt, 2010. Relational Mathematics. Cambridge University Press, ISBN 978-0-521-76268-7.