Interval order

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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[edit]

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]


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 the OEIS)