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 $(2+2)$ 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 $P=(X,\leq )$ 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 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 $P$ 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 $[n]$ 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 $[2n]$, a left nesting is an $i\in [2n]$ such that $i<i+1<f(i+1)<f(i)$ and a right nesting is an $i\in [2n]$ such that $f(i)<f(i+1)<i<i+1$.

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

$F(t)=\sum _{n\geq 0}\prod _{i=1}^{n}(1-(1-t)^{i}).$

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 (sequence A022493 in the OEIS) begins

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

References

• Bousquet-Mélou, Mireille; Claesson, Anders; Dukes, Mark; Kitaev, Sergey (2010), "(2+2) free posets, ascent sequences and pattern avoiding permutations", Journal of Combinatorial Theory, Series A, 117 (7): 884–909, arXiv:0806.0666, doi:10.1016/j.jcta.2009.12.007, MR 2652101, S2CID 8677150.
• Felsner, S. (1992), Interval Orders: Combinatorial Structure and Algorithms (PDF), Ph.D. dissertation, Technical University of Berlin.
• Felsner, S.; Habib, M.; Möhring, R. H. (1994), "On the interplay between interval dimension and dimension" (PDF), SIAM Journal on Discrete Mathematics, 7 (1): 32–40, doi:10.1137/S089548019121885X, MR 1259007.
• Fishburn, Peter C. (1970), "Intransitive indifference with unequal indifference intervals", Journal of Mathematical Psychology, 7 (1): 144–149, doi:10.1016/0022-2496(70)90062-3, MR 0253942.
• Zagier, Don (2001), "Vassiliev invariants and a strange identity related to the Dedekind eta-function", Topology, 40 (5): 945–960, doi:10.1016/s0040-9383(00)00005-7, MR 1860536.

Further reading

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