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, I₁, being considered less than another, I₂, if I₁ is completely to the left of I₂. More formally, a 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, the $(2+2)$ free posets .^[1]

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 containment orders on intervals on the real line (equivalently, the orders of dimension ≤ 2).

Interval dimension

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]

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 $2n$ .^[4] These are the involutions with no left or right neighbor nestings where, for $f$ an involution 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 ^[5]

$F(t) = \sum_{n \geq 0} \prod_{i = 1}^n (1-(1-t)^i)$.

Hence the number of unlabeled interval orders of size $n$ is given by the coefficient of $t^n$ in the expansion of $F(t)$.

1, 2, 5, 15, 53, 217, 1014, 5335, 31240, 201608, 1422074, 10886503, 89903100, 796713190, 7541889195, 75955177642, … (sequence A022493 in OEIS)

Notes

1. Fishburn (1970)
2. http://page.math.tu-berlin.de/~felsner/Paper/Idim-dim.pdf p.2
3. http://page.math.tu-berlin.de/~felsner/Paper/diss.pdf, p.47
4. Bousquet-Mélou et al. (2010)
5. Zagier (2001)

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 (Academic Press, Inc.) 117 (7): 884–909, doi:10.1016/j.jcta.2009.12.007, ISSN 0097-3165
• 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, ISSN 0040-9383
• Fishburn, Peter (1985), Interval Orders and Interval Graphs: A Study of Partially Ordered Sets, John Wiley
• 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.

• Gunther Schmidt, 2010. Relational Mathematics. Cambridge University Press, ISBN 978-0-521-76268-7.