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 countable poset is an interval order if and only if there exists a bijection.

Such posets may be equivalently characterized as those with no induced subposet isomorphic to the pair of two-element chains-free posets..

, is precisely the semiorders.

The complement of the comparability graph, ≤) is the interval graph.

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 archaeological histories of pottery styles.

Interval orders and dimension

An important parameter of partial orders is order dimension is the least number of linear orders. 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 is the least integer. The interval dimension of an order is never greater than its order dimension.

Combinatorics

are also in bijection with a subset of fixed-point-free involutions on ordered sets with cardinality..

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

. The sequence of these numbers begins

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

Notes

References

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Further reading