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, ''I''₁, being considered less than another, ''I''₂, if ''I''₁ is completely to the left of ''I''₂.
More formally, a countable poset $P = (X, \leq)$ is an interval order if and only if
there exists a bijection from $X$ to a set of real intervals,
so $x_i \mapsto (\ell_i, r_i)$,
such that for any $x_i, x_j \in X$ we have
$x_i < x_j$ in $P$ exactly when $r_i < \ell_j$.

Such posets may be equivalently
characterized as those with no induced subposet isomorphic to the
pair of two-element chains, in other words as the $(2+2)$-free posets
. Fully written out, this means that for any two pairs of elements $a > b$ and $c > d$ one must have $a > d$ or $c > b$.

The subclass of interval orders obtained by restricting the intervals to those of unit length, so they all have the form $(\ell_i, \ell_i + 1)$, is precisely the semiorders.

The complement of the comparability graph of an interval order ($X$, ≤)
is the interval graph $(X, \cap)$.

Interval orders should not be confused with the interval-containment orders, which are the inclusion orders on intervals on the real line (equivalently, the orders of dimension ≤ 2).

Interval orders' practical applications include modelling evolution of species and archeological histories of pottery styles.

Interval orders and dimension

An important parameter of partial orders is order dimension: the dimension of a partial order $P$ is the least number of linear orders whose intersection is $P$. For interval orders, dimension can be arbitrarily large. And while the problem of determining the dimension of general partial orders is known to be NP-hard, determining the dimension of an interval order remains a problem of unknown computational complexity.

A related parameter is interval dimension, which is defined analogously, but in terms of interval orders instead of linear orders. Thus, the interval dimension of a partially ordered set $P = (X, \leq)$ is the least integer $k$ for which there exist interval orders $\preceq_1, \ldots, \preceq_k$ on $X$ with $x \leq y$ exactly when $x \preceq_1 y, \ldots,$ and $x \preceq_k y$.
The interval dimension of an order is never greater than its order dimension.

Combinatorics

In addition to being isomorphic to $(2+2)$-free posets,
unlabeled interval orders on /math> are also in bijection
with a subset of fixed-point-free involutions
on ordered sets with cardinality $2n$
. These are the
involutions with no so-called left- or right-neighbor nestings where, for any involution
$f$ on n/math>, a left nesting is
an i \in n/math> such that $i < i+1 < f(i+1) < f(i)$ and a right nesting is an i \in n/math> such that
$f(i) < f(i+1) < i < i+1$.

Such involutions, according to semi-length, have ordinary generating function

: $F(t) = \sum_ \prod_^n (1-(1-t)^i).$

The coefficient of $t^n$ in the expansion of $F(t)$ gives the number of unlabeled interval orders of size $n$. The sequence of these numbers begins

:1, 2, 5, 15, 53, 217, 1014, 5335, 31240, 201608, 1422074, 10886503, 89903100, 796713190, 7541889195, 75955177642, …

Notes

References

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Further reading

*{{citation
, last = Fishburn
, first = Peter , author-link = Peter C. Fishburn
, title = Interval Orders and Interval Graphs: A Study of Partially Ordered Sets
, publisher = John Wiley
, year = 1985

Order theory
Combinatorics