Interval order

A partial order on the set {a, b, c, d, e, f} illustrated by its Hasse diagram (left) and a collection of intervals that represents it (right).

The [math]\displaystyle{ (2+2) }[/math] poset (black Hasse diagram) cannot be part of an interval order: if a is completely right of b, and d overlaps with both a and b, and c is completely right of d, then c must be completely right of b (light gray edge).

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 [math]\displaystyle{ P = (X, \leq) }[/math] is an interval order if and only if there exists a bijection from [math]\displaystyle{ X }[/math] to a set of real intervals, so [math]\displaystyle{ x_i \mapsto (\ell_i, r_i) }[/math], such that for any [math]\displaystyle{ x_i, x_j \in X }[/math] we have [math]\displaystyle{ x_i \lt x_j }[/math] in [math]\displaystyle{ P }[/math] exactly when [math]\displaystyle{ r_i \lt \ell_j }[/math].

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 [math]\displaystyle{ (2+2) }[/math]-free posets . Fully written out, this means that for any two pairs of elements [math]\displaystyle{ a \gt b }[/math] and [math]\displaystyle{ c \gt d }[/math] one must have [math]\displaystyle{ a \gt d }[/math] or [math]\displaystyle{ c \gt b }[/math].

The subclass of interval orders obtained by restricting the intervals to those of unit length, so they all have the form [math]\displaystyle{ (\ell_i, \ell_i + 1) }[/math], is precisely the semiorders.

The complement of the comparability graph of an interval order ([math]\displaystyle{ X }[/math], ≤) is the interval graph [math]\displaystyle{ (X, \cap) }[/math].

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 and dimension

Unsolved problem in mathematics: What is the complexity of determining the order dimension of an interval order? (more unsolved problems in mathematics)

An important parameter of partial orders is order dimension: the dimension of a partial order [math]\displaystyle{ P }[/math] is the least number of linear orders whose intersection is [math]\displaystyle{ P }[/math]. 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 [math]\displaystyle{ P = (X, \leq) }[/math] is the least integer [math]\displaystyle{ k }[/math] for which there exist interval orders [math]\displaystyle{ \preceq_1, \ldots, \preceq_k }[/math] on [math]\displaystyle{ X }[/math] with [math]\displaystyle{ x \leq y }[/math] exactly when [math]\displaystyle{ x \preceq_1 y, \ldots, }[/math] and [math]\displaystyle{ x \preceq_k y }[/math]. The interval dimension of an order is never greater than its order dimension.

Combinatorics

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

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

[math]\displaystyle{ F(t) = \sum_{n \geq 0} \prod_{i = 1}^n (1-(1-t)^i). }[/math]

The coefficient of [math]\displaystyle{ t^n }[/math] in the expansion of [math]\displaystyle{ F(t) }[/math] gives the number of unlabeled interval orders of size [math]\displaystyle{ n }[/math]. The sequence of these numbers (sequence A022493 in the OEIS) begins

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

References

• "(2+2) free posets, ascent sequences and pattern avoiding permutations", Journal of Combinatorial Theory, Series A 117 (7): 884–909, 2010, doi:10.1016/j.jcta.2009.12.007 .
• Felsner, S. (1992), Interval Orders: Combinatorial Structure and Algorithms, Ph.D. dissertation, Technical University of Berlin, http://page.math.tu-berlin.de/~felsner/Paper/diss.pdf .
• Felsner, S.; Habib, M.; Möhring, R. H. (1994), "On the interplay between interval dimension and dimension", SIAM Journal on Discrete Mathematics 7 (1): 32–40, doi:10.1137/S089548019121885X, http://page.math.tu-berlin.de/~felsner/Paper/Idim-dim.pdf .
• "Intransitive indifference with unequal indifference intervals", Journal of Mathematical Psychology 7 (1): 144–149, 1970, doi:10.1016/0022-2496(70)90062-3 .
• "Vassiliev invariants and a strange identity related to the Dedekind eta-function", Topology 40 (5): 945–960, 2001, doi:10.1016/s0040-9383(00)00005-7 .

Further reading

• Fishburn (1985), Interval Orders and Interval Graphs: A Study of Partially Ordered Sets, John Wiley